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Group Theory

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Contents Articles History of group theory

1

Group (mathematics)

7

Group theory

27

Elementary group theory

34

Symmetry group

40

Symmetric group

44

Combinatorial group theory

53

Algebraic group

54

Solvable group

56

Solvable subgroup

59

Tits building

62

Finite group

67

p-adic number

69

Tits alternative

76

Finitely generated group

77

Linear group

79

Finite index

81

Free subgroup

85

Tits group

88

Tits–Koecher construction

90

Primitive group

91

Geometric group theory

92

Hyperbolic group

98

Automatic group

101

Discrete group

103

Todd–Coxeter algorithm

105

Frobenius group

107

Zassenhaus group

109

Regular p-group

110

Isoclinism of groups

111

Variety (universal algebra)

113

Reflection group

115

Fundamental group

117

Classical group

122

Unitary group

124

Character theory

128

Sylow theorem

133

Lie algebra

139

Class group

144

Abelian group

148

Lie group

155

Galois group

164

General linear group

165

Representation theory

170

Symmetry in physics

181

Space group

186

Molecular symmetry

193

Applications of group theory

198

Examples of groups

205

Modular representation theory

210

Conway group

215

Mathieu group

219

Sporadic groups

230

Janko group J1

234

Janko group J2

237

Janko group J3

239

Janko group J4

240

Fischer group

241

Baby Monster group

243

Monster group

244

References Article Sources and Contributors

248

Image Sources, Licenses and Contributors

252

Article Licenses License

253

History of group theory

1

History of group theory The history of group theory, a mathematical domain studying groups in their various forms, has evolved in various parallel threads. There are three historical roots of group theory: the theory of algebraic equations, number theory and geometry.[1] [2] [3] Lagrange, Abel and Galois were early researchers in the field of group theory.

Early 19th century The earliest study of groups as such probably goes back to the work of Lagrange in the late 18th century. However, this work was somewhat isolated, and 1846 publications of Cauchy and Galois are more commonly referred to as the beginning of group theory. The theory did not develop in a vacuum, and so 3 important threads in its pre-history are developed here.

Development of permutation groups One foundational root of group theory was the quest of solutions of polynomial equations of degree higher than 4. An early source occurs in the problem of forming an equation of degree m having as its roots m of the roots of a given equation of degree n > m. For simple cases the problem goes back to Hudde (1659). Saunderson (1740) noted that the determination of the quadratic factors of a biquadratic expression necessarily leads to a sextic equation, and Le Sœur (1748) and Waring (1762 to 1782) still further elaborated the idea.[3] A common foundation for the theory of equations on the basis of the group of permutations was found by mathematician Lagrange (1770, 1771), and on this was built the theory of substitutions. He discovered that the roots of all resolvents (résolvantes, réduites) which he examined are rational functions of the roots of the respective equations. To study the properties of these functions he invented a Calcul des Combinaisons. The contemporary work of Vandermonde (1770) also foreshadowed the coming theory.[3] Ruffini (1799) attempted a proof of the impossibility of solving the quintic and higher equations. Ruffini distinguished what are now called intransitive and transitive, and imprimitive and primitive groups, and (1801) uses the group of an equation under the name l'assieme delle permutazioni. He also published a letter from Abbati to himself, in which the group idea is prominent.[3] Galois found that if r1, r2, ... rn are the n roots of an equation, there is always a group of permutations of the r's such that • every function of the roots invariable by the substitutions of the group is rationally known, and • conversely, every rationally determinable function of the roots is invariant under the substitutions of the group. In modern terms, the solvability of the Galois group attached to the equation determines the solvability of the equation with radicals. Galois also contributed to the theory of modular equations and to that of elliptic functions. His first publication on group theory was made at the age of eighteen (1829), but his contributions attracted little attention until the publication of his collected papers in 1846 (Liouville, Vol. XI).[4] [5] Galois is honored as the first mathematician linking group theory and field theory, with the theory that is now called Galois theory.[3]

Galois age fifteen, drawn by a classmate.

Groups similar to Galois groups are (today) called permutation groups, a concept investigated in particular by Cauchy. A number of important theorems in early group theory is due to Cauchy. Cayley's On the theory of groups, as depending on the symbolic equation θn = 1 (1854) gives the first abstract definition of finite groups.

History of group theory

2

Groups related to geometry Secondly, the systematic use of groups in geometry, mainly in the guise of symmetry groups, was initiated by Klein's 1872 Erlangen program.[6] The study of what are now called Lie groups started systematically in 1884 with Sophus Lie, followed by work of Killing, Study, Schur, Maurer, and Cartan. The discontinuous (discrete group) theory was built up by Felix Klein, Lie, Poincaré, and Charles Émile Picard, in connection in particular with modular forms and monodromy.

Felix Klein

Sophus Lie

History of group theory

3

Appearance of groups in number theory The third root of group theory was number theory. Certain abelian group structures had been implicitly used in number-theoretical work by Gauss, and more explicitly by Kronecker.[7] Early attempts to prove Fermat's last theorem were led to a climax by Kummer by introducing groups describing factorization into prime numbers.[8]

Ernst Kummer

Convergence Group theory as an increasingly independent subject was popularized by Serret, who devoted section IV of his algebra to the theory; by Camille Jordan, whose Traité des substitutions et des équations algébriques (1870) is a classic; and to Eugen Netto (1882), whose Theory of Substitutions and its Applications to Algebra was translated into English by Cole (1892). Other group theorists of the nineteenth century were Bertrand, Charles Hermite, Frobenius, Leopold Kronecker, and Émile Mathieu;[3] as well as Burnside, Dickson, Hölder, Moore, Sylow, and Weber.

Camille Jordan

The convergence of the above three sources into a uniform theory started with Jordan's Traité and von Dyck (1882) who first defined a group in the full modern sense. The textbooks of Weber and Burnside helped establish group theory as a discipline.[9] The abstract group formulation did not apply to a large portion of 19th century group theory, and an alternative formalism was given in terms of Lie algebras.

Late 19th century Groups in the 1870-1900 period were described as the continuous groups of Lie, the discontinuous groups, finite groups of substitutions of roots (gradually being called permutations), and finite groups of linear substitutions (usually of finite fields). During the 1880-1920 period, groups described by presentations came into a life of their own through the work of Arthur Cayley, Walther von Dyck, Dehn, Nielsen, Schreier, and continued in the 1920-1940 period with the work of Coxeter, Magnus, and others to form the field of combinatorial group theory. Finite groups in the 1870-1900 period saw such highlights as the Sylow theorems, Hölder's classification of groups of square-free order, and the early beginnings of the character theory of Frobenius. Already by 1860, the groups of automorphisms of the finite projective planes had been studied (by Mathieu), and in the 1870s Felix Klein's group-theoretic vision of geometry was being realized in his Erlangen program. The automorphism groups of higher

History of group theory dimensional projective spaces were studied by Jordan in his Traité and included composition series for most of the so called classical groups, though he avoided non-prime fields and omitted the unitary groups. The study was continued by Moore and Burnside, and brought into comprehensive textbook form by Leonard Dickson in 1901. The role of simple groups was emphasized by Jordan, and criteria for non-simplicity were developed by Hölder until he was able to classify the simple groups of order less than 200. The study was continued by F. N. Cole (up to 660) and Burnside (up to 1092), and finally in an early "millennium project", up to 2001 by Miller and Ling in 1900. Continuous groups in the 1870-1900 period developed rapidly. Killing and Lie's foundational papers were published, Hilbert's theorem in invariant theory 1882, etc.

Early 20th century In the period 1900-1940, infinite "discontinuous" (now called discrete groups) groups gained life of their own. Burnside's famous problem ushered in the study of arbitrary subgroups of finite dimensional linear groups over arbitrary fields, and indeed arbitrary groups. Fundamental groups and reflection groups encouraged the developments of J. A. Todd and Coxeter, such as the Todd–Coxeter algorithm in combinatorial group theory. Algebraic groups, defined as solutions of polynomial equations (rather than acting on them, as in the earlier century), benefited heavily from the continuous theory of Lie. Neumann and Neumann produced their study of varieties of groups, groups defined by group theoretic equations rather than polynomial ones. Continuous groups also had explosive growth in the 1900-1940 period. Topological groups began to be studied as such. There were many great achievements in continuous groups: Cartan's classification of semisimple Lie algebras, Weyl's theory of representations of compact groups, Haar's work in the locally compact case. Finite groups in the 1900-1940 grew immensely. This period witnessed the birth of character theory by Frobenius, Burnside, and Schur which helped answer many of the 19th century questions in permutation groups, and opened the way to entirely new techniques in abstract finite groups. This period saw the work of Hall: on a generalization of Sylow's theorem to arbitrary sets of primes which revolutionized the study of finite soluble groups, and on the power-commutator structure of p-groups, including the ideas of regular p-groups and isoclinism of groups, which revolutionized the study of p-groups and was the first major result in this area since Sylow. This period saw Zassenhaus's famous Schur-Zassenhaus theorem on the existence of complements to Hall's generalization of Sylow subgroups, as well as his progress on Frobenius groups, and a near classification of Zassenhaus groups.

Mid 20th century Both depth, breadth and also the impact of group theory subsequently grew. The domain started branching out into areas such as algebraic groups, group extensions, and representation theory.[10] Starting in the 1950s, in a huge collaborative effort, group theorists succeeded to classify all finite simple groups in 1982. Completing and simplifying the proof of the classification are areas of active research.[11] Anatoly Maltsev also made important contributions to group theory during this time; his early work was in logic in the 1930s, but in the 1940s he proved important embedding properties of semigroups into groups, studied the isomorphism problem of group rings, established the Malçev correspondence for polycyclic groups, and in the 1960s return to logic proving various theories within the study of groups to be undecidable. Earlier, Alfred Tarski proved elementary group theory undecidable.[12]

4

History of group theory

Later 20th century The period of 1960-1980 was one of excitement in many areas of group theory. In finite groups, there were many independent milestones. One had the discovery of 22 new sporadic groups, and the completion of the first generation of the classification of finite simple groups. One had the influential idea of the Carter subgroup, and the subsequent creation of formation theory and the theory of classes of groups. One had the remarkable extensions of Clifford theory by Green to the indecomposable modules of group algebras. During this era, the field of computational group theory became a recognized field of study, due in part to its tremendous success during the first generation classification. In discrete groups, the geometric methods of Tits and the availability the surjectivity of Lang's map allowed a revolution in algebraic groups. The Burnside problem had tremendous progress, with better counterexamples constructed in the 60s and early 80s, but the finishing touches "for all but finitely many" were not completed until the 90s. The work on the Burnside problem increased interest in Lie algebras in exponent p, and the methods of Lazard began to see a wider impact, especially in the study of p-groups. Continuous groups broadened considerably, with p-adic analytic questions becoming important. Many conjectures were made during this time, including the coclass conjectures.

Late 20th century The last twenty years of the twentieth century enjoyed the successes of over one hundred years of study in group theory. In finite groups, post classification results included the O'Nan–Scott theorem, the Aschbacher classification, the classification of multiply transitive finite groups, the determination of the maximal subgroups of the simple groups and the corresponding classifications of primitive groups. In finite geometry and combinatorics, many problems could now be settled. The modular representation theory entered a new era as the techniques of the classification were axiomatized, including fusion systems, Puig's theory of pairs and nilpotent blocks. The theory of finite soluble groups was likewise transformed by the influential book of Doerk–Hawkes which brought the theory of projectors and injectors to a wider audience. In discrete groups, several areas of geometry came together to produce exciting new fields. Work on knot theory, orbifolds, hyperbolic manifolds, and groups acting on trees (the Bass–Serre theory), much enlivened the study of hyperbolic groups, automatic groups. Questions such as Thurston's 1982 geometrization conjecture, inspired entirely new techniques in geometric group theory and low dimensional topology, and was involved in the solution of one of the Millennium Prize Problems, the Poincaré conjecture. Continuous groups saw the solution of the problem of hearing the shape of a drum in 1992 using symmetry groups of the laplacian operator. Continuous techniques were applied to many aspects of group theory using function spaces and quantum groups. Many 18th and 19th century problems are now revisited in this more general setting, and many questions in the theory of the representations of groups have answers.

5

History of group theory

Today Group theory continues to be an intensely studied matter. Its importance to contemporary mathematics as a whole may be seen from the 2008 Abel Prize, awarded to John Griggs Thompson and Jacques Tits for their contributions to group theory.

Notes [1] [2] [3] [4] [5] [6] [7] [8] [9]

Wussing 2007 Kleiner 1986 Smith 1906 Galois 1908 Kleiner 1986, p. 202 Wussing 2007, §III.2 Kleiner 1986, p. 204 Wussing 2007, §I.3.4 Solomon writes in Burnside's Collected Works, "The effect of [Burnside's book] was broader and more pervasive, influencing the entire course of non-commutative algebra in the twentieth century." [10] Curtis 2003 [11] Aschbacher 2004 [12] Tarski, Alfred (1953) "Undecidability of the elementary theory of groups" in Tarski, Mostowski, and Raphael Robinson Undecidable Theories. North-Holland: 77-87.

References • Historically important publications in group theory. • Curtis, Charles W. (2003), Pioneers of Representation Theory: Frobenius, Burnside, Schur, and Brauer, History of Mathematics, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-2677-5 • Galois, Évariste (1908), Tannery, Jules, ed., Manuscrits de Évariste Galois (http://quod.lib.umich.edu/cgi/t/ text/text-idx?c=umhistmath;idno=AAN9280), Paris: Gauthier-Villars • Kleiner, Israel (1986), "The evolution of group theory: a brief survey" (http://www.jstor.org/ sici?sici=0025-570X(198610)59:4<195:TEOGTA>2.0.CO;2-9), Mathematics Magazine 59 (4): 195–215, doi:10.2307/2690312, MR863090, ISSN 0025-570X • Smith, David Eugene (1906), History of Modern Mathematics (http://www.gutenberg.org/etext/8746), Mathematical Monographs, No. 1 • Wussing, Hans (2007), The Genesis of the Abstract Group Concept: A Contribution to the History of the Origin of Abstract Group Theory, New York: Dover Publications, ISBN 978-0-486-45868-7 • du Sautoy, Marcus (2008), Finding Moonshine, London: Fourth Estate, ISBN 978-0-00-721461-7

6

Group (mathematics)

7

Group (mathematics) In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity and invertibility. Many familiar mathematical structures such as number systems obey these axioms: for example, the integers endowed with the addition operation form a group. However, the abstract formalization of the group axioms, detached as it is from the concrete nature of any particular group and its operation, allows entities with highly diverse mathematical origins in abstract algebra and beyond to be handled in a flexible way, while retaining their essential structural aspects. The ubiquity of groups in numerous areas within and outside mathematics makes them a central organizing principle of contemporary mathematics.[1] [2]

The possible manipulations of this Rubik's Cube form a group.

Groups share a fundamental kinship with the notion of symmetry. A symmetry group encodes symmetry features of a geometrical object: it consists of the set of transformations that leave the object unchanged, and the operation of combining two such transformations by performing one after the other. Such symmetry groups, particularly the continuous Lie groups, play an important role in many academic disciplines. Matrix groups, for example, can be used to understand fundamental physical laws underlying special relativity and symmetry phenomena in molecular chemistry. The concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s. After contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Modern group theory—a very active mathematical discipline—studies groups in their own right.a[›] To explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups. In addition to their abstract properties, group theorists also study the different ways in which a group can be expressed concretely (its group representations), both from a theoretical and a computational point of view. A particularly rich theory has been developed for finite groups, which culminated with the monumental classification of finite simple groups completed in 1983. Since the mid-1980s, geometric group theory, which studies finitely generated groups as geometric objects, has become a particularly active area in group theory.

Definition and illustration First example: the integers One of the most familiar groups is the set of integers Z which consists of the numbers ..., −4, −3, −2, −1, 0, 1, 2, 3, 4, ...[3] The following properties of integer addition serve as a model for the abstract group axioms given in the definition below. 1. For any two integers a and b, the sum a + b is also an integer. In other words, the process of adding integers two at a time always yields an integer, not some other type of number such as a fraction. This property is known as closure under addition. 2. For all integers a, b and c, (a + b) + c = a + (b + c). Expressed in words, adding a to b first, and then adding the result to c gives the same final result as adding a to the sum of b and c, a property known as associativity.

Group (mathematics) 3. If a is any integer, then 0 + a = a + 0 = a. Zero is called the identity element of addition because adding it to any integer returns the same integer. 4. For every integer a, there is an integer b such that a + b = b + a = 0. The integer b is called the inverse element of the integer a and is denoted −a. The integers, together with the operation +, form a mathematical object belonging to a broad class sharing similar structural aspects. To appropriately understand these structures as a collective, the following abstract definition is developed.

Definition A group is a set, G, together with an operation • (called the group law of G) that combines any two elements a and b to form another element, denoted a • b or ab. To qualify as a group, the set and operation, (G, •), must satisfy four requirements known as the group axioms:[4] Closure For all a, b in G, the result of the operation, a • b, is also in G.b[›] Associativity For all a, b and c in G, (a • b) • c = a • (b • c). Identity element There exists an element e in G, such that for every element a in G, the equation e • a = a • e = a holds. The identity element of a group G is often written as 1 or 1G,[5] a notation inherited from the multiplicative identity. Inverse element For each a in G, there exists an element b in G such that a • b = b • a = 1G. The order in which the group operation is carried out can be significant. In other words, the result of combining element a with element b need not yield the same result as combining element b with element a; the equation a•b=b•a may not always be true. This equation does always hold in the group of integers under addition, because a + b = b + a for any two integers (commutativity of addition). However, it does not always hold in the symmetry group below. Groups for which the equation a • b = b • a always holds are called abelian (in honor of Niels Abel). Thus, the integer addition group is abelian, but the following symmetry group is not. The set G is called the underlying set of the group (G, •). Often the group's underlying set G is used as a short name for the group (G, •). Along the same lines, sometimes a shorthand expression such as "a subset of the group G" is used when what is actually meant is "a subset of the underlying set G of the group (G, •)." Usually, it is clear from the context whether a symbol like G refers to a group or to an underlying set.

Second example: a symmetry group The symmetries (i.e., rotations and reflections) of a square form a group called a dihedral group, and denoted D4.[6] The following symmetries occur:

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Group (mathematics)

9

id (keeping it as is)

r1 (rotation by 90° right)

r2 (rotation by 180° right)

r3 (rotation by 270° right)

fv (vertical flip)

fh (horizontal flip)

fd (diagonal flip)

fc (counter-diagonal flip)

The elements of the symmetry group of the square (D4). The vertices are colored and numbered only to visualize the operations.

• the identity operation leaving everything unchanged, denoted id; • rotations of the square by 90° right, 180° right, and 270° right, denoted by r1, r2 and r3, respectively; • reflections about the vertical and horizontal middle line (fh and fv), or through the two diagonals (fd and fc). The defining operation of this group is function composition: The eight symmetries are functions from the square to the square, and two symmetries are combined by composing them as functions, that is, applying them to the square one at a time. The result of performing first a and then b is written symbolically from right to left as b • a ("apply the symmetry b after performing the symmetry a"). The right-to-left notation is the same notation that is used for composition of functions. The group table on the right lists the results of all such compositions possible. For example, rotating by 270° right (r3) and then flipping horizontally (fh) is the same as performing a reflection along the diagonal (fd). Using the above symbols, highlighted in blue in the group table: fh • r3 = fd.

Group table of D4 •

id

r1

r2

r3

fv

fh

fd

fc

id

id

r1

r2

r3

fv

fh

fd

fc

r1

r1

r2

r3

id

fc

fd

fv

fh

r2

r2

r3

id

r1

fh

fv

fc

fd

r3

r3

id

r1

r2

fd

fc

fh

fv

fv

fv

fd

fh

fc

id

r2

r1

r3

fh

fh

fc

fv

fd

r2

id

r3

r1

fd

fd

fh

fc

fv

r3

r1

id

r2

fc

fc

fv

fd

fh

r1

r3

r2

id

The elements id, r1, r2, and r3 form a subgroup, highlighted in red (upper left region). A left and right coset of this subgroup is highlighted in green (in the last row) and yellow (last column), respectively.

Given this set of symmetries and the described operation, the group axioms can be understood as follows: 1. The closure axiom demands that the composition b • a of any two symmetries a and b is also a symmetry. Another example for the group operation is r3 • fh = fc,

Group (mathematics)

10

i.e. rotating 270° right after flipping horizontally equals flipping along the counter-diagonal (fc). Indeed every other combination of two symmetries still gives a symmetry, as can be checked using the group table. 2. The associativity constraint deals with composing more than two symmetries: Starting with three elements a, b and c of D4, there are two possible ways of using these three symmetries in this order to determine a symmetry of the square. One of these ways is to first compose a and b into a single symmetry, then to compose that symmetry with c. The other way is to first compose b and c, then to compose the resulting symmetry with a. The associativity condition (a • b) • c = a • (b • c) means that these two ways are the same, i.e., a product of many group elements can be simplified in any order. For example, (fd • fv) • r2 = fd • (fv • r2) can be checked using the group table at the right (fd • fv) • r2 = r3 • r2 = r1, which equals fd • (fv • r2) = fd • fh =

r1.

While associativity is true for the symmetries of the square and addition of numbers, it is not true for all operations. For instance, subtraction of numbers is not associative: (7 − 3) − 2 = 2 is not the same as 7 − (3 − 2) = 6. 3. The identity element is the symmetry id leaving everything unchanged: for any symmetry a, performing id after a (or a after id) equals a, in symbolic form, id • a = a, a • id = a. 4. An inverse element undoes the transformation of some other element. Every symmetry can be undone: each of transformations—identity id, the flips fh, fv, fd, fc and the 180° rotation r2—is its own inverse, because performing each one twice brings the square back to its original orientation. The rotations r3 and r1 are each other's inverse, because rotating 90° and then rotation 270° (or vice versa) yields a rotation over 360° which leaves the square unchanged. In symbols, fh • fh = id, r3 • r1 = r1 • r3 = id. In contrast to the group of integers above, where the order of the operation is irrelevant, it does matter in D4: fh • r1 = fc but r1 • fh = fd. In other words, D4 is not abelian, which makes the group structure more difficult than the integers introduced first.

History The modern concept of an abstract group developed out of several fields of mathematics.[7] [8] [9] The original motivation for group theory was the quest for solutions of polynomial equations of degree higher than 4. The 19th-century French mathematician Évariste Galois, extending prior work of Paolo Ruffini and Joseph-Louis Lagrange, gave a criterion for the solvability of a particular polynomial equation in terms of the symmetry group of its roots (solutions). The elements of such a Galois group correspond to certain permutations of the roots. At first, Galois' ideas were rejected by his contemporaries, and published only posthumously.[10] [11] More general permutation groups were investigated in particular by Augustin Louis Cauchy. Arthur Cayley's On the theory of groups, as depending on the symbolic equation θn = 1 (1854) gives the first abstract definition of a finite group.[12] Geometry was a second field in which groups were used systematically, especially symmetry groups as part of Felix Klein's 1872 Erlangen program.[13] After novel geometries such as hyperbolic and projective geometry had emerged, Klein used group theory to organize them in a more coherent way. Further advancing these ideas, Sophus Lie founded the study of Lie groups in 1884.[14]

Group (mathematics) The third field contributing to group theory was number theory. Certain abelian group structures had been used implicitly in Carl Friedrich Gauss' number-theoretical work Disquisitiones Arithmeticae (1798), and more explicitly by Leopold Kronecker.[15] In 1847, Ernst Kummer led early attempts to prove Fermat's Last Theorem to a climax by developing groups describing factorization into prime numbers.[16] The convergence of these various sources into a uniform theory of groups started with Camille Jordan's Traité des substitutions et des équations algébriques (1870).[17] Walther von Dyck (1882) gave the first statement of the modern definition of an abstract group.[18] As of the 20th century, groups gained wide recognition by the pioneering work of Ferdinand Georg Frobenius and William Burnside, who worked on representation theory of finite groups, Richard Brauer's modular representation theory and Issai Schur's papers.[19] The theory of Lie groups, and more generally locally compact groups was pushed by Hermann Weyl, Élie Cartan and many others.[20] Its algebraic counterpart, the theory of algebraic groups, was first shaped by Claude Chevalley (from the late 1930s) and later by pivotal work of Armand Borel and Jacques Tits.[21] The University of Chicago's 1960–61 Group Theory Year brought together group theorists such as Daniel Gorenstein, John G. Thompson and Walter Feit, laying the foundation of a collaboration that, with input from numerous other mathematicians, classified all finite simple groups in 1982. This project exceeded previous mathematical endeavours by its sheer size, in both length of proof and number of researchers. Research is ongoing to simplify the proof of this classification.[22] These days, group theory is still a highly active mathematical branch crucially impacting many other fields.a[›]

Elementary consequences of the group axioms Basic facts about all groups that can be obtained directly from the group axioms are commonly subsumed under elementary group theory.[23] For example, repeated applications of the associativity axiom show that the unambiguity of a • b • c = (a • b) • c = a • (b • c) generalizes to more than three factors. Because this implies that parentheses can be inserted anywhere within such a series of terms, parentheses are usually omitted.[24] The axioms may be weakened to assert only the existence of a left identity and left inverses. Both can be shown to be actually two-sided, so the resulting definition is equivalent to the one given above.[25]

Uniqueness of identity element and inverses Two important consequences of the group axioms are the uniqueness of the identity element and the uniqueness of inverse elements. There can be only one identity element in a group, and each element in a group has exactly one inverse element. Thus, it is customary to speak of the identity, and the inverse of an element.[26] To prove the uniqueness of an inverse element of a, suppose that a has two inverses, denoted l and r, in a group (G, •). Then

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Group (mathematics)

12

l = l • 1G

as 1G is the identity element

= l • (a • r) because r is an inverse of a, so 1G = a • r = (l • a) • r by associativity, which allows to rearrange the parentheses = 1G • r

since l is an inverse of a, i.e. l • a = 1G

= r

for 1G is the identity element

The two extremal terms l and r are equal, since they are connected by a chain of equalities. In other words there is only one inverse element of a. Similarly, to prove that the identity element of a group is unique, assume G is a group with two identity elements 1G and e. Then 1G = 1G • e = e, hence 1G and e are equal.

Division In groups, it is possible to perform division: given elements a and b of the group G, there is exactly one solution x in G to the equation x • a = b.[26] In fact, right multiplication of the equation by a−1 gives the solution x = x • a • a−1 = b • a−1. Similarly there is exactly one solution y in G to the equation a • y = b, namely y = a−1 • b. In general, x and y need not agree. A consequence of this is that multiplying by a group element g is a bijection. Specifically, if g is an element of the group G, there is a bijection from G to itself called left translation by g sending h ∈ G to g • h. Similarly, right translation by g is a bijection from G to itself sending h to h • g. If G is abelian, left and right translation by a group element are the same.

Basic concepts To understand groups beyond the level of mere symbolic manipulations as above, more structural concepts have to be employed.c[›] There is a conceptual principle underlying all of the following notions: to take advantage of the structure offered by groups (which sets, being "structureless", do not have), constructions related to groups have to be compatible with the group operation. This compatibility manifests itself in the following notions in various ways. For example, groups can be related to each other via functions called group homomorphisms. By the mentioned principle, they are required to respect the group structures in a precise sense. The structure of groups can also be understood by breaking them into pieces called subgroups and quotient groups. The principle of "preserving structures"—a recurring topic in mathematics throughout—is an instance of working in a category, in this case the category of groups.[27]

Group homomorphisms Group homomorphismsg[›] are functions that preserve group structure. A function a: G → H between two groups (G,•) and (H,*) is a homomorphism if the equation a(g • k) = a(g) * a(k) holds for all elements g, k in G. In other words, the result is the same when performing the group operation after or before applying the map a. This requirement ensures that a(1G) = 1H, and also a(g)−1 = a(g−1) for all g in G. Thus a group homomorphism respects all the structure of G provided by the group axioms.[28] Two groups G and H are called isomorphic if there exist group homomorphisms a: G → H and b: H → G, such that applying the two functions one after another (in each of the two possible orders) equal the identity function of G and H, respectively. That is, a(b(h)) = h and b(a(g)) = g for any g in G and h in H. From an abstract point of view, isomorphic groups carry the same information. For example, proving that g • g = 1G for some element g of G is equivalent to proving that a(g) • a(g) = 1H, because applying a to the first equality yields the second, and applying b to the second gives back the first.

Group (mathematics)

Subgroups Informally, a subgroup is a group H contained within a bigger one, G.[29] Concretely, the identity element of G is contained in H, and whenever h1 and h2 are in H, then so are h1 • h2 and h1−1, so the elements of H, equipped with the group operation on G restricted to H, form indeed a group. In the example above, the identity and the rotations constitute a subgroup R = {id, r1, r2, r3}, highlighted in red in the group table above: any two rotations composed are still a rotation, and a rotation can be undone by (i.e. is inverse to) the complementary rotations 270° for 90°, 180° for 180°, and 90° for 270° (note that rotation in the opposite direction is not defined). The subgroup test is a necessary and sufficient condition for a subset H of a group G to be a subgroup: it is sufficient to check that g−1h ∈ H for all elements g, h ∈ H. Knowing the subgroups is important in understanding the group as a whole.d[›] Given any subset S of a group G, the subgroup generated by S consists of products of elements of S and their inverses. It is the smallest subgroup of G containing S.[30] In the introductory example above, the subgroup generated by r2 and fv consists of these two elements, the identity element id and fh = fv • r2. Again, this is a subgroup, because combining any two of these four elements or their inverses (which are, in this particular case, these same elements) yields an element of this subgroup.

Cosets In many situations it is desirable to consider two group elements the same if they differ by an element of a given subgroup. For example, in D4 above, once a flip is performed, the square never gets back to the r2 configuration by just applying the rotation operations (and no further flips), i.e. the rotation operations are irrelevant to the question whether a flip has been performed. Cosets are used to formalize this insight: a subgroup H defines left and right cosets, which can be thought of as translations of H by arbitrary group elements g. In symbolic terms, the left and right coset of H containing g are gH = {g • h, h ∈ H} and Hg = {h • g, h ∈ H}, respectively.[31] The cosets of any subgroup H form a partition of G; that is, the union of all left cosets is equal to G and two left cosets are either equal or have an empty intersection.[32] The first case g1H = g2H happens precisely when g1−1 • g2 ∈ H, i.e. if the two elements differ by an element of H. Similar considerations apply to the right cosets of H. The left and right cosets of H may or may not be equal. If they are, i.e. for all g in G, gH = Hg, then H is said to be a normal subgroup. One may then simply refer to N as the set of cosets. In D4, the introductory symmetry group, the left cosets gR of the subgroup R consisting of the rotations are either equal to R, if g is an element of R itself, or otherwise equal to U = fcR = {fc, fv, fd, fh} (highlighted in green). The subgroup R is also normal, because fcR = U = Rfc and similarly for any element other than fc.

Quotient groups In addition to disregarding the internal structure of a subgroup by considering its cosets, it is desirable to endow this coarser entity with a group law called quotient group or factor group. For this to be possible, the subgroup has to be normal. Given any normal subgroup N, the quotient group is defined by G / N = {gN, g ∈ G}, "G modulo N".[33] This set inherits a group operation (sometimes called coset multiplication, or coset addition) from the original group G: (gN) • (hN) = (gh)N for all g and h in G. This definition is motivated by the idea (itself an instance of general structural considerations outlined above) that the map G → G / N that associates to any element g its coset gN be a group homomorphism, or by general abstract considerations called universal properties. The coset eN = N serves as the identity in this group, and the inverse of gN in the quotient group is (gN)−1 = (g−1)N.e[›]

13

Group (mathematics)

14

•

R

U

R

R

U

U

U

R

Group table of the quotient group D4 / R.

The elements of the quotient group D4 / R are R itself, which represents the identity, and U = fvR. The group operation on the quotient is shown at the right. For example, U • U = fvR • fvR = (fv • fv)R = R. Both the subgroup R = {id, r1, r2, r3}, as well as the corresponding quotient are abelian, whereas D4 is not abelian. Building bigger groups by smaller ones, such as D4 from its subgroup R and the quotient D4 / R is abstracted by a notion called semidirect product. Quotient and subgroups together form a way of describing every group by its presentation: any group is the quotient of the free group over the generators of the group, quotiented by the subgroup of relations. The dihedral group D4, for example, can be generated by two elements r and f (for example, r = r1, the right rotation and f = fv the vertical (or any other) flip), which means that every symmetry of the square is a finite composition of these two symmetries or their inverses. Together with the relations r 4 = f 2 = (r • f)2 = 1,[34] the group is completely described. A presentation of a group can also be used to construct the Cayley graph, a device used to graphically capture discrete groups. Sub- and quotient groups are related in the following way: a subset H of G can be seen as an injective map H → G, i.e. any element of the target has at most one element that maps to it. The counterpart to injective maps are surjective maps (every element of the target is mapped onto), such as the canonical map G → G / N.y[›] Interpreting subgroup and quotients in light of these homomorphisms emphasizes the structural concept inherent to these definitions alluded to in the introduction. In general, homomorphisms are neither injective nor surjective. Kernel and image of group homomorphisms and the first isomorphism theorem address this phenomenon.

Examples and applications

A periodic wallpaper pattern gives rise to a wallpaper group.

The fundamental group of a plane minus a point (bold) consists of loops around the missing point. This group is isomorphic to the integers.

Group (mathematics)

Examples and applications of groups abound. A starting point is the group Z of integers with addition as group operation, introduced above. If instead of addition multiplication is considered, one obtains multiplicative groups. These groups are predecessors of important constructions in abstract algebra. Groups are also applied in many other mathematical areas. Mathematical objects are often examined by associating groups to them and studying the properties of the corresponding groups. For example, Henri Poincaré founded what is now called algebraic topology by introducing the fundamental group.[35] By means of this connection, topological properties such as proximity and continuity translate into properties of groups.i[›] For example, elements of the fundamental group are represented by loops. The second image at the right shows some loops in a plane minus a point. The blue loop is considered null-homotopic (and thus irrelevant), because it can be continuously shrunk to a point. The presence of the hole prevents the orange loop from being shrunk to a point. The fundamental group of the plane with a point deleted turns out to be infinite cyclic, generated by the orange loop (or any other loop winding once around the hole). This way, the fundamental group detects the hole. In more recent applications, the influence has also been reversed to motivate geometric constructions by a group-theoretical background.j[›] In a similar vein, geometric group theory employs geometric concepts, for example in the study of hyperbolic groups.[36] Further branches crucially applying groups include algebraic geometry and number theory.[37] In addition to the above theoretical applications, many practical applications of groups exist. Cryptography relies on the combination of the abstract group theory approach together with algorithmical knowledge obtained in computational group theory, in particular when implemented for finite groups.[38] Applications of group theory are not restricted to mathematics; sciences such as physics, chemistry and computer science benefit from the concept.

Numbers Many number systems, such as the integers and the rationals enjoy a naturally given group structure. In some cases, such as with the rationals, both addition and multiplication operations give rise to group structures. Such number systems are predecessors to more general algebraic structures known as rings and fields. Further abstract algebraic concepts such as modules, vector spaces and algebras also form groups. Integers The group of integers Z under addition, denoted (Z, +), has been described above. The integers, with the operation of multiplication instead of addition, (Z, ·) do not form a group. The closure, associativity and identity axioms are satisfied, but inverses do not exist: for example, a = 2 is an integer, but the only solution to the equation a · b = 1 in this case is b = 1/2, which is a rational number, but not an integer. Hence not every element of Z has a (multiplicative) inverse.k[›] Rationals The desire for the existence of multiplicative inverses suggests considering fractions

Fractions of integers (with b nonzero) are known as rational numbers.l[›] The set of all such fractions is commonly denoted Q. There is still a minor obstacle for (Q, ·), the rationals with multiplication, being a group: because the rational number 0 does not have a multiplicative inverse (i.e., there is no x such that x · 0 = 1), (Q, ·) is still not a group. However, the set of all nonzero rational numbers Q \ {0} = {q ∈ Q, q ≠ 0} does form an abelian group under multiplication, denoted (Q \ {0}, ·).m[›] Associativity and identity element axioms follow from the properties of integers. The closure requirement still holds true after removing zero, because the product of two nonzero rationals is never zero. Finally, the inverse of a/b is b/a, therefore the axiom of the inverse element is satisfied.

15

Group (mathematics)

16

The rational numbers (including 0) also form a group under addition. Intertwining addition and multiplication operations yields more complicated structures called rings and—if division is possible, such as in Q—fields, which occupy a central position in abstract algebra. Group theoretic arguments therefore underlie parts of the theory of those entities.n[›] Nonzero integers modulo a prime For any prime number p, modular arithmetic furnishes the multiplicative group of integers modulo p.[39] Its elements are integers not divisible by p, considered modulo p, i.e. two numbers are considered equivalent if their difference is divisible by p. For example, if p = 5, there are exactly four group elements 1, 2, 3, 4: multiples of 5 are excluded and 6 and −4 are both equivalent to 1 etc. The group operation is given by multiplication. Therefore, 4 · 4 = 1, because the usual product 16 is equivalent to 1, for 5 divides 16 − 1 = 15, denoted 16 ≡ 1 (mod 5). The primality of p ensures that the product of two integers neither of which is divisible by p is not divisible by p either, hence the indicated set of classes is closed under multiplication.o[›] The identity element is 1, as usual for a multiplicative group, and the associativity follows from the corresponding property of integers. Finally, the inverse element axiom requires that given an integer a not divisible by p, there exists an integer b such that a · b ≡ 1 (mod p), i.e. p divides the difference a · b − 1. The inverse b can be found by using Bézout's identity and the fact that the greatest common divisor gcd(a, p) equals 1.[40] In the case p = 5 above, the inverse of 4 is 4, and the inverse of 3 is 2, as 3 · 2 = 6 ≡ 1 (mod 5). Hence all group axioms are fulfilled. Actually, this example is similar to (Q\{0}, ·) above, because it turns out to be the multiplicative group of nonzero elements in the finite field Fp, denoted Fp×.[41] These groups are crucial to public-key cryptography.p[›]

Cyclic groups A cyclic group is a group all of whose elements are powers (when the group operation is written additively, the term 'multiple' can be used) of a particular element a.[42] In multiplicative notation, the elements of the group are: ..., a−3, a−2, a−1, a0 = e, a, a2, a3, ..., where a2 means a • a, and a−3 stands for a−1 • a−1 • a−1=(a • a • a)−1 etc.h[›] Such an element a is called a generator or a primitive element of the group. A typical example for this class of groups is the group of n-th complex roots of unity, given by complex numbers z satisfying zn = 1 (and whose operation is multiplication).[43] Any cyclic group with n elements is isomorphic to this group. Using some field theory, the group Fp× can be shown to be cyclic: for example, if p = 5, 3 is a generator since 31 = 3, 32 = 9 ≡ 4, 33 ≡ 2, and 34 ≡ 1.

The 6th complex roots of unity form a cyclic group. z is a primitive element, but z2 is not, because the odd powers of z are not a power of z2.

Some cyclic groups have an infinite number of elements. In these groups, for every non-zero element a, all the powers of a are distinct; despite the name "cyclic group", the powers of the elements do not cycle. An infinite cyclic group is isomorphic to (Z, +), the group of integers under addition introduced above.[44] As these two prototypes are both abelian, so is any cyclic group. The study of abelian groups is quite mature, including the fundamental theorem of finitely generated abelian groups; and reflecting this state of affairs, many group-related notions, such as center and commutator, describe the extent to which a given group is not abelian.[45]

Group (mathematics)

17

Symmetry groups Symmetry groups are groups consisting of symmetries of given mathematical objects—be they of geometric nature, such as the introductory symmetry group of the square, or of algebraic nature, such as polynomial equations and their solutions.[46] Conceptually, group theory can be thought of as the study of symmetry.t[›] Symmetries in mathematics greatly simplify the study of geometrical or analytical objects. A group is said to act on another mathematical object X if every group element performs some operation on X compatibly to the group law. In the rightmost example below, an element of order 7 of the (2,3,7) triangle group acts on the tiling by permuting the highlighted warped triangles (and the other ones, too). By a group action, the group pattern is connected to the structure of the object being acted on. In chemical fields, such as crystallography, space groups and point groups describe molecular symmetries and crystal symmetries. These symmetries underlie the chemical and physical behavior of these systems, and group theory enables simplification of quantum mechanical analysis of these properties.[47] For example, group theory is used to show that optical transitions between certain quantum levels cannot occur simply because of the symmetry of the states involved. Not only are groups useful to assess the implications of symmetries in molecules, but surprisingly they also predict that molecules sometimes can change symmetry. The Jahn-Teller effect is a distortion of a molecule of high symmetry when it adopts a particular ground state of lower symmetry from a set of possible ground states that are related to each other by the symmetry operations of the molecule.[48] [49]

Rotations and flips form the symmetry group of a great icosahedron.

Likewise, group theory helps predict the changes in physical properties that occur when a material undergoes a phase transition, for example, from a cubic to a tetrahedral crystalline form. An example is ferroelectric materials, where the change from a paraelectric to a ferroelectric state occurs at the Curie temperature and is related to a change from the high-symmetry paraelectric state to the lower symmetry ferroelectic state, accompanied by a so-called soft phonon mode, a vibrational lattice mode that goes to zero frequency at the transition.[50] Such spontaneous symmetry breaking has found further application in elementary particle physics, where its occurrence is related to the appearance of Goldstone bosons.

Buckminsterfullerene Ammonia, NH3. Its displays symmetry group is of order 6, icosahedral symmetry. generated by a 120° rotation and a reflection.

Cubane C8H8 features octahedral symmetry.

Hexaaquacopper(II) complex ion, [Cu(OH2)6]2+. Compared to a perfectly symmetrical shape, the molecule is vertically dilated by about 22% (Jahn-Teller effect).

The (2,3,7) triangle group, a hyperbolic group, acts on this tiling of the hyperbolic plane.

Finite symmetry groups such as the Mathieu groups are used in coding theory, which is in turn applied in error correction of transmitted data, and in CD players.[51] Another application is differential Galois theory, which

Group (mathematics)

18

characterizes functions having antiderivatives of a prescribed form, giving group-theoretic criteria for when solutions of certain differential equations are well-behaved.u[›] Geometric properties that remain stable under group actions are investigated in (geometric) invariant theory.[52]

General linear group and representation theory Matrix groups consist of matrices together with matrix multiplication. The general linear group GL(n, R) consists of all invertible n-by-n matrices with real entries.[53] Its subgroups are referred to as matrix groups or linear groups. The dihedral group example mentioned above can be viewed as a (very small) matrix group. Another important matrix group is the special orthogonal group SO(n). It describes all possible rotations in n dimensions. Via Euler angles, rotation matrices are used in computer graphics.[54]

Two vectors (the left illustration) multiplied by matrices (the middle and right illustrations). The middle illustration represents a clockwise rotation by 90°, while the right-most one stretches the x-coordinate by factor 2.

Representation theory is both an application of the group concept and important for a deeper understanding of groups.[55] [56] It studies the group by its group actions on other spaces. A broad class of group representations are linear representations, i.e. the group is acting on a vector space, such as the three-dimensional Euclidean space R3. A representation of G on an n-dimensional real vector space is simply a group homomorphism ρ: G → GL(n, R) from the group to the general linear group. This way, the group operation, which may be abstractly given, translates to the multiplication of matrices making it accessible to explicit computations.w[›] Given a group action, this gives further means to study the object being acted on.x[›] On the other hand, it also yields information about the group. Group representations are an organizing principle in the theory of finite groups, Lie groups, algebraic groups and topological groups, especially (locally) compact groups.[55] [57]

Galois groups Galois groups have been developed to help solve polynomial equations by capturing their symmetry features.[58] [59] For example, the solutions of the quadratic equation ax2 + bx + c = 0 are given by

Exchanging "+" and "−" in the expression, i.e. permuting the two solutions of the equation can be viewed as a (very simple) group operation. Similar formulae are known for cubic and quartic equations, but do not exist in general for degree 5 and higher.[60] Abstract properties of Galois groups associated with polynomials (in particular their solvability) give a criterion for polynomials that have all their solutions expressible by radicals, i.e. solutions expressible using solely addition, multiplication, and roots similar to the formula above.[61] The problem can be dealt with by shifting to field theory and considering the splitting field of a polynomial. Modern Galois theory generalizes the above type of Galois groups to field extensions and establishes—via the fundamental theorem of Galois theory—a precise relationship between fields and groups, underlining once again the ubiquity of groups in mathematics.

Group (mathematics)

Finite groups A group is called finite if it has a finite number of elements. The number of elements is called the order of the group G.[62] An important class is the symmetric groups SN, the groups of permutations of N letters. For example, the symmetric group on 3 letters S3 is the group consisting of all possible swaps of the three letters ABC, i.e. contains the elements ABC, ACB, ..., up to CBA, in total 6 (or 3 factorial) elements. This class is fundamental insofar as any finite group can be expressed as a subgroup of a symmetric group SN for a suitable integer N (Cayley's theorem). Parallel to the group of symmetries of the square above, S3 can also be interpreted as the group of symmetries of an equilateral triangle. The order of an element a in a group G is the least positive integer n such that a n = e, where a n represents

i.e. application of the operation • to n copies of a. (If • represents multiplication, then an corresponds to the nth power of a.) In infinite groups, such an n may not exist, in which case the order of a is said to be infinity. The order of an element equals the order of the cyclic subgroup generated by this element. More sophisticated counting techniques, for example counting cosets, yield more precise statements about finite groups: Lagrange's Theorem states that for a finite group G the order of any finite subgroup H divides the order of G. The Sylow theorems give a partial converse. The dihedral group (discussed above) is a finite group of order 8. The order of r1 is 4, as is the order of the subgroup R it generates (see above). The order of the reflection elements fv etc. is 2. Both orders divide 8, as predicted by Lagrange's Theorem. The groups Fp× above have order p − 1.

Classification of finite simple groups Mathematicians often strive for a complete classification (or list) of a mathematical notion. In the context of finite groups, this aim quickly leads to difficult and profound mathematics. According to Lagrange's theorem, finite groups of order p, a prime number, are necessarily cyclic (abelian) groups Zp. Groups of order p2 can also be shown to be abelian, a statement which does not generalize to order p3, as the non-abelian group D4 of order 8 = 23 above shows.[63] Computer algebra systems can be used to list small groups, but there is no classification of all finite groups.q[›] An intermediate step is the classification of finite simple groups.r[›] A nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself.s[›] The Jordan–Hölder theorem exhibits finite simple groups as the building blocks for all finite groups.[64] Listing all finite simple groups was a major achievement in contemporary group theory. 1998 Fields Medal winner Richard Borcherds succeeded to prove the monstrous moonshine conjectures, a surprising and deep relation of the largest finite simple sporadic group—the "monster group"—with certain modular functions, a piece of classical complex analysis, and string theory, a theory supposed to unify the description of many physical phenomena.[65]

19

Group (mathematics)

20

Groups with additional structure Many groups are simultaneously groups and examples of other mathematical structures. In the language of category theory, they are group objects in a category, meaning that they are objects (that is, examples of another mathematical structure) which come with transformations (called morphisms) that mimic the group axioms. For example, every group (as defined above) is also a set, so a group is a group object in the category of sets.

Topological groups Some topological spaces may be endowed with a group law. In order for the group law and the topology to interweave well, the group operations must be continuous functions, that is, g • h, and g−1 must not vary wildly if g and h vary only little. Such groups are called topological groups, and they are the group objects in the category of topological spaces.[66] The most basic examples are the reals R under addition, (R \ {0}, ·), and similarly with any other topological field such as the complex numbers or p-adic numbers. All of these groups are locally compact, so they have Haar measures and can be studied via harmonic analysis. The former offer an abstract formalism of invariant integrals. Invariance means, in the case of real numbers for example:

The unit circle in the complex plane under complex multiplication is a Lie group and, therefore, a topological group. It is topological since complex multiplication and division are continuous. It is a manifold and thus a Lie group, because every small piece, such as the red arc in the figure, looks like a part of the real line (shown at the bottom).

for any constant c. Matrix groups over these fields fall under this regime, as do adele rings and adelic algebraic groups, which are basic to number theory.[67] Galois groups of infinite field extensions such as the absolute Galois group can also be equipped with a topology, the so-called Krull topology, which in turn is central to generalize the above sketched connection of fields and groups to infinite field extensions.[68] An advanced generalization of this idea, adapted to the needs of algebraic geometry, is the étale fundamental group.[69]

Lie groups Lie groups (in honor of Sophus Lie) are groups which also have a manifold structure, i.e. they are spaces looking locally like some Euclidean space of the appropriate dimension.[70] Again, the additional structure, here the manifold structure, has to be compatible, i.e. the maps corresponding to multiplication and the inverse have to be smooth. A standard example is the general linear group introduced above: it is an open subset of the space of all n-by-n matrices, because it is given by the inequality det (A) ≠ 0, where A denotes an n-by-n matrix.[71] Lie groups are of fundamental importance in physics: Noether's theorem links continuous symmetries to conserved quantities.[72] Rotation, as well as translations in space and time are basic symmetries of the laws of mechanics.

Group (mathematics)

21

They can, for instance, be used to construct simple models—imposing, say, axial symmetry on a situation will typically lead to significant simplification in the equations one needs to solve to provide a physical description.v[›] Another example are the Lorentz transformations, which relate measurements of time and velocity of two observers in motion relative to each other. They can be deduced in a purely group-theoretical way, by expressing the transformations as a rotational symmetry of Minkowski space. The latter serves—in the absence of significant gravitation—as a model of space time in special relativity.[73] The full symmetry group of Minkowski space, i.e. including translations, is known as the Poincaré group. By the above, it plays a pivotal role in special relativity and, by implication, for quantum field theories.[74] Symmetries that vary with location are central to the modern description of physical interactions with the help of gauge theory.[75]

Generalizations Group-like structures Totality Associativity Identity Inverses Group

Yes

Yes

Yes

Yes

Monoid

Yes

Yes

Yes

No

Semigroup

Yes

Yes

No

No

Loop

Yes

No

Yes

Yes

Quasigroup

Yes

No

No

Yes

Magma

Yes

No

No

No

Groupoid

No

Yes

Yes

Yes

Category

No

Yes

Yes

No

In abstract algebra, more general structures are defined by relaxing some of the axioms defining a group.[27] [76] [77] For example, if the requirement that every element has an inverse is eliminated, the resulting algebraic structure is called a monoid. The natural numbers N (including 0) under addition form a monoid, as do the nonzero integers under multiplication (Z \ {0}, ·), see above. There is a general method to formally add inverses to elements to any (abelian) monoid, much the same way as (Q \ {0}, ·) is derived from (Z \ {0}, ·), known as the Grothendieck group. Groupoids are similar to groups except that the composition a • b need not be defined for all a and b. They arise in the study of more complicated forms of symmetry, often in topological and analytical structures, such as the fundamental groupoid or stacks. Finally, it is possible to generalize any of these concepts by replacing the binary operation with an arbitrary n-ary one (i.e. an operation taking n arguments). With the proper generalization of the group axioms this gives rise to an n-ary group.[78] The table gives a list of several structures generalizing groups.

Notes ^ a: Mathematical Reviews lists 3,224 research papers on group theory and its generalizations written in 2005. ^ b: The closure axiom is already implied by the condition that • be a binary operation. Some authors therefore omit this axiom. Lang 2002 ^ c: See, for example, the books of Lang (2002, 2005) and Herstein (1996, 1975). ^ d: However, a group is not determined by its lattice of subgroups. See Suzuki 1951. ^ e: The fact that the group operation extends this canonically is an instance of a universal property. ^ f: For example, if G is finite, then the size of any subgroup and any quotient group divides the size of G, according to Lagrange's theorem. ^ g: The word homomorphism derives from Greek ὁμός—the same and μορφή—structure. ^ h: The additive notation for elements of a cyclic group would be t • a, t in Z.

Group (mathematics) ^ i: See the Seifert–van Kampen theorem for an example. ^ j: An example is group cohomology of a group which equals the singular homology of its classifying space. ^ k: Elements which do have multiplicative inverses are called units, see Lang 2002, §II.1, p. 84. ^ l: The transition from the integers to the rationals by adding fractions is generalized by the quotient field. ^ m: The same is true for any field F instead of Q. See Lang 2005, §III.1, p. 86. ^ n: For example, a finite subgroup of the multiplicative group of a field is necessarily cyclic. See Lang 2002, Theorem IV.1.9. The notions of torsion of a module and simple algebras are other instances of this principle. ^ o: The stated property is a possible definition of prime numbers. See prime element. ^ p: For example, the Diffie-Hellman protocol uses the discrete logarithm. ^ q: The groups of order at most 2000 are known. Up to isomorphism, there are about 49 billion. See Besche, Eick & O'Brien 2001. ^ r: The gap between the classification of simple groups and the one of all groups lies in the extension problem, a problem too hard to be solved in general. See Aschbacher 2004, p. 737. ^ s: Equivalently, a nontrivial group is simple if its only quotient groups are the trivial group and the group itself. See Michler 2006, Carter 1989. ^ t: More rigorously, every group is the symmetry group of some graph; see Frucht's theorem, Frucht 1939. ^ u: More precisely, the monodromy action on the vector space of solutions of the differential equations is considered. See Kuga 1993, pp. 105–113. ^ v: See Schwarzschild metric for an example where symmetry greatly reduces the complexity of physical systems. ^ w: This was crucial to the classification of finite simple groups, for example. See Aschbacher 2004. ^ x: See, for example, Schur's Lemma for the impact of a group action on simple modules. A more involved example is the action of an absolute Galois group on étale cohomology. ^ y: Injective and surjective maps correspond to mono- and epimorphisms, respectively. They are interchanged when passing to the dual category.

Citations [1] Herstein 1975, §2, p. 26 [2] Hall 1967, §1.1, p. 1: "The idea of a group is one which pervades the whole of mathematics both pure and applied." [3] Lang 2005, App. 2, p. 360 [4] Herstein 1975, §2.1, p. 27 [5] Weisstein, Eric W., " Identity Element (http:/ / mathworld. wolfram. com/ IdentityElement. html)" from MathWorld. [6] Herstein 1975, §2.6, p. 54 [7] Wussing 2007 [8] Kleiner 1986 [9] Smith 1906 [10] Galois 1908 [11] Kleiner 1986, p. 202 [12] Cayley 1889 [13] Wussing 2007, §III.2 [14] Lie 1973 [15] Kleiner 1986, p. 204 [16] Wussing 2007, §I.3.4 [17] Jordan 1870 [18] von Dyck 1882 [19] Curtis 2003 [20] Mackey 1976 [21] Borel 2001 [22] Aschbacher 2004 [23] Ledermann 1953, §1.2, pp. 4–5 [24] Ledermann 1973, §I.1, p. 3 [25] Lang 2002, §I.2, p. 7 [26] Lang 2005, §II.1, p. 17

22

Group (mathematics) [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48]

Mac Lane 1998 Lang 2005, §II.3, p. 34 Lang 2005, §II.1, p. 19 Ledermann 1973, §II.12, p. 39 Lang 2005, §II.4, p. 41 Lang 2002, §I.2, p. 12 Lang 2005, §II.4, p. 45 Lang 2002, §I.2, p. 9 Hatcher 2002, Chapter I, p. 30 Coornaert, Delzant & Papadopoulos 1990 for example, class groups and Picard groups; see Neukirch 1999, in particular §§I.12 and I.13 Seress 1997 Lang 2005, Chapter VII Rosen 2000, p. 54 (Theorem 2.1) Lang 2005, §VIII.1, p. 292 Lang 2005, §II.1, p. 22 Lang 2005, §II.2, p. 26 Lang 2005, §II.1, p. 22 (example 11) Lang 2002, §I.5, p. 26, 29 Weyl 1952 Conway, Delgado Friedrichs & Huson et al. 2001. See also Bishop 1993 Bersuker, Isaac (2006), The Jahn-Teller Effect, Cambridge University Press, p. 2, ISBN 0521822122

[49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78]

Jahn & Teller 1937 Dove, Martin T (2003), Structure and Dynamics: an atomic view of materials, Oxford University Press, p. 265, ISBN 0198506783 Welsh 1989 Mumford, Fogarty & Kirwan 1994 Lay 2003 Kuipers 1999 Fulton & Harris 1991 Serre 1977 Rudin 1990 Robinson 1996, p. viii Artin 1998 Lang 2002, Chapter VI (see in particular p. 273 for concrete examples) Lang 2002, p. 292 (Theorem VI.7.2) Kurzweil & Stellmacher 2004 Artin 1991, Theorem 6.1.14. See also Lang 2002, p. 77 for similar results. Lang 2002, §I. 3, p. 22 Ronan 2007 Husain 1966 Neukirch 1999 Shatz 1972 Milne 1980 Warner 1983 Borel 1991 Goldstein 1980 Weinberg 1972 Naber 2003 Becchi 1997 Denecke & Wismath 2002 Romanowska & Smith 2002 Dudek 2001

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Group (mathematics)

References General references • Artin, Michael (1991), Algebra, Prentice Hall, ISBN 978-0-89871-510-1, Chapter 2 contains an undergraduate-level exposition of the notions covered in this article. • Devlin, Keith (2000), The Language of Mathematics: Making the Invisible Visible, Owl Books, ISBN 978-0-8050-7254-9, Chapter 5 provides a layman-accessible explanation of groups. • Fulton, William; Harris, Joe (1991), Representation theory. A first course, Graduate Texts in Mathematics, Readings in Mathematics, 129, New York: Springer-Verlag, MR1153249, ISBN 978-0-387-97527-6, ISBN 978-0-387-97495-8. • Hall, G. G. (1967), Applied group theory, American Elsevier Publishing Co., Inc., New York, MR0219593, an elementary introduction. • Herstein, Israel Nathan (1996), Abstract algebra (3rd ed.), Upper Saddle River, NJ: Prentice Hall Inc., MR1375019, ISBN 978-0-13-374562-7. • Herstein, Israel Nathan (1975), Topics in algebra (2nd ed.), Lexington, Mass.: Xerox College Publishing, MR0356988. • Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, 211 (Revised third ed.), New York: Springer-Verlag, MR1878556, ISBN 978-0-387-95385-4 • Lang, Serge (2005), Undergraduate Algebra (3rd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-22025-3. • Ledermann, Walter (1953), Introduction to the theory of finite groups, Oliver and Boyd, Edinburgh and London, MR0054593. • Ledermann, Walter (1973), Introduction to group theory, New York: Barnes and Noble, OCLC 795613. • Robinson, Derek John Scott (1996), A course in the theory of groups, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94461-6.

Special references • Artin, Emil (1998), Galois Theory, New York: Dover Publications, ISBN 978-0-486-62342-9. • Aschbacher, Michael (2004), "The Status of the Classification of the Finite Simple Groups" (http://www.ams. org/notices/200407/fea-aschbacher.pdf) (PDF), Notices of the American Mathematical Society 51 (7): 736–740, ISSN 0002-9920. • Becchi, C. (1997), Introduction to Gauge Theories (http://www.arxiv.org/abs/hep-ph/9705211), retrieved 2008-05-15. • Besche, Hans Ulrich; Eick, Bettina; O'Brien, E. A. (2001), "The groups of order at most 2000" (http://www. ams.org/era/2001-07-01/S1079-6762-01-00087-7/home.html), Electronic Research Announcements of the American Mathematical Society 7: 1–4, doi:10.1090/S1079-6762-01-00087-7, MR1826989. • Bishop, David H. L. (1993), Group theory and chemistry, New York: Dover Publications, ISBN 978-0-486-67355-4. • Borel, Armand (1991), Linear algebraic groups, Graduate Texts in Mathematics, 126 (2nd ed.), Berlin, New York: Springer-Verlag, MR1102012, ISBN 978-0-387-97370-8. • Carter, Roger W. (1989), Simple groups of Lie type, New York: John Wiley & Sons, ISBN 978-0-471-50683-6. • Conway, John Horton; Delgado Friedrichs, Olaf; Huson, Daniel H.; Thurston, William P. (2001), "On three-dimensional space groups" (http://arxiv.org/abs/math.MG/9911185), Beiträge zur Algebra und Geometrie 42 (2): 475–507, MR1865535, ISSN 0138-4821. • (French) Coornaert, M.; Delzant, T.; Papadopoulos, A. (1990), Géométrie et théorie des groupes [Geometry and Group Theory], Lecture Notes in Mathematics, 1441, Berlin, New York: Springer-Verlag, MR1075994, ISBN 978-3-540-52977-4.

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Group (mathematics) • Denecke, Klaus; Wismath, Shelly L. (2002), Universal algebra and applications in theoretical computer science, London: CRC Press, ISBN 978-1-58488-254-1. • Dudek, W.A. (2001), "On some old problems in n-ary groups" (http://www.quasigroups.eu/contents/ contents8.php?m=trzeci), Quasigroups and Related Systems 8: 15–36. • (German) Frucht, R. (1939), "Herstellung von Graphen mit vorgegebener abstrakter Gruppe [Construction of Graphs with Prescribed Group (http://www.numdam.org/numdam-bin/fitem?id=CM_1939__6__239_0)"], Compositio Mathematica 6: 239–50, ISSN 0010-437X. • Goldstein, Herbert (1980), Classical Mechanics (2nd ed.), Reading, MA: Addison-Wesley Publishing, pp. 588–596, ISBN 0-201-02918-9. • Hatcher, Allen (2002), Algebraic topology (http://www.math.cornell.edu/~hatcher/AT/ATpage.html), Cambridge University Press, ISBN 978-0-521-79540-1. • Husain, Taqdir (1966), Introduction to Topological Groups, Philadelphia: W.B. Saunders Company, ISBN 978-0-89874-193-3 • Jahn, H.; Teller, E. (1937), "Stability of Polyatomic Molecules in Degenerate Electronic States. I. Orbital Degeneracy", Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences (1934–1990) 161 (905): 220–235, doi:10.1098/rspa.1937.0142. • Kuipers, Jack B. (1999), Quaternions and rotation sequences—A primer with applications to orbits, aerospace, and virtual reality, Princeton University Press, MR1670862, ISBN 978-0-691-05872-6. • Kuga, Michio (1993), Galois' dream: group theory and differential equations, Boston, MA: Birkhäuser Boston, MR1199112, ISBN 978-0-8176-3688-3. • Kurzweil, Hans; Stellmacher, Bernd (2004), The theory of finite groups, Universitext, Berlin, New York: Springer-Verlag, MR2014408, ISBN 978-0-387-40510-0. • Lay, David (2003), Linear Algebra and Its Applications, Addison-Wesley, ISBN 978-0-201-70970-4. • Mac Lane, Saunders (1998), Categories for the Working Mathematician (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-98403-2. • Michler, Gerhard (2006), Theory of finite simple groups, Cambridge University Press, ISBN 978-0-521-86625-5. • Milne, James S. (1980), Étale cohomology, Princeton University Press, ISBN 978-0-691-08238-7 • Mumford, David; Fogarty, J.; Kirwan, F. (1994), Geometric invariant theory, 34 (3rd ed.), Berlin, New York: Springer-Verlag, MR1304906, ISBN 978-3-540-56963-3. • Naber, Gregory L. (2003), The geometry of Minkowski spacetime, New York: Dover Publications, MR2044239, ISBN 978-0-486-43235-9. • Neukirch, Jürgen (1999), Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften, 322, Berlin: Springer-Verlag, MR1697859, ISBN 978-3-540-65399-8. • Romanowska, A.B.; Smith, J.D.H. (2002), Modes, World Scientific, ISBN 9789810249427. • Ronan, Mark (2007), Symmetry and the Monster: The Story of One of the Greatest Quests of Mathematics, Oxford University Press, ISBN 978-0-19-280723-6. • Rosen, Kenneth H. (2000), Elementary number theory and its applications (4th ed.), Addison-Wesley, MR1739433, ISBN 978-0-201-87073-2. • Rudin, Walter (1990), Fourier Analysis on Groups, Wiley Classics, Wiley-Blackwell, ISBN 047152364X. • Seress, Ákos (1997), "An introduction to computational group theory" (http://www.math.ohio-state.edu/ ~akos/notices.ps), Notices of the American Mathematical Society 44 (6): 671–679, MR1452069, ISSN 0002-9920. • Serre, Jean-Pierre (1977), Linear representations of finite groups, Berlin, New York: Springer-Verlag, MR0450380, ISBN 978-0-387-90190-9. • Shatz, Stephen S. (1972), Profinite groups, arithmetic, and geometry, Princeton University Press, MR0347778, ISBN 978-0-691-08017-8

25

Group (mathematics) • Suzuki, Michio (1951), "On the lattice of subgroups of finite groups" (http://jstor.org/stable/1990375), Transactions of the American Mathematical Society 70 (2): 345–371, doi:10.2307/1990375. • Warner, Frank (1983), Foundations of Differentiable Manifolds and Lie Groups, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90894-6. • Weinberg, Steven (1972), Gravitation and Cosmology, New York: John Wiley & Sons, ISBN 0-471-92567-5. • Welsh, Dominic (1989), Codes and cryptography, Oxford: Clarendon Press, ISBN 978-0-19-853287-3. • Weyl, Hermann (1952), Symmetry, Princeton University Press, ISBN 978-0-691-02374-8.

Historical references • Borel, Armand (2001), Essays in the History of Lie Groups and Algebraic Groups, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0288-5 • Cayley, Arthur (1889), The collected mathematical papers of Arthur Cayley (http://www.hti.umich.edu/cgi/t/ text/pageviewer-idx?c=umhistmath;cc=umhistmath;rgn=full text;idno=ABS3153.0001.001;didno=ABS3153. 0001.001;view=image;seq=00000140), II (1851–1860), Cambridge University Press. • O'Connor, J.J; Robertson, E.F. (1996), The development of group theory (http://www-groups.dcs.st-and.ac.uk/ ~history/HistTopics/Development_group_theory.html). • Curtis, Charles W. (2003), Pioneers of Representation Theory: Frobenius, Burnside, Schur, and Brauer, History of Mathematics, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-2677-5. • (German) von Dyck, Walther (1882), "Gruppentheoretische Studien (Group-theoretical Studies)" (http://www. springerlink.com/content/t8lx644qm87p3731) (subscription required), Mathematische Annalen 20 (1): 1–44, doi:10.1007/BF01443322, ISSN 0025-5831. • (French) Galois, Évariste (1908), Tannery, Jules, ed., Manuscrits de Évariste Galois [Évariste Galois' Manuscripts (http://quod.lib.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=AAN9280)], Paris: Gauthier-Villars (Galois work was first published by Joseph Liouville in 1843). • (French) Jordan, Camille (1870), Traité des substitutions et des équations algébriques [Study of Substitutions and Algebraic Equations (http://gallica.bnf.fr/notice?N=FRBNF35001297)], Paris: Gauthier-Villars. • Kleiner, Israel (1986), "The evolution of group theory: a brief survey" (http://www.jstor.org/ sici?sici=0025-570X(198610)59:4<195:TEOGTA>2.0.CO;2-9) (subscription required), Mathematics Magazine 59 (4): 195–215, doi:10.2307/2690312, MR863090, ISSN 0025-570X. • (German) Lie, Sophus (1973), Gesammelte Abhandlungen. Band 1 [Collected papers. Volume 1], New York: Johnson Reprint Corp., MR0392459. • Mackey, George Whitelaw (1976), The theory of unitary group representations, University of Chicago Press, MR0396826 • Smith, David Eugene (1906), History of Modern Mathematics (http://www.gutenberg.org/etext/8746), Mathematical Monographs, No. 1. • Wussing, Hans (2007), The Genesis of the Abstract Group Concept: A Contribution to the History of the Origin of Abstract Group Theory, New York: Dover Publications, ISBN 978-0-486-45868-7.

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Group theory In mathematics and abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have strongly influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced tremendous advances and have become subject areas in their own right. Various physical systems, such as crystals and the hydrogen atom, can be modelled by symmetry groups. Thus group theory and the closely related representation theory have many applications in physics and chemistry. One of the most important mathematical achievements of the 20th century was the collaborative effort, taking up more than 10,000 journal pages and mostly published between 1960 and 1980, that culminated in a complete classification of finite simple groups.

History Group theory has three main historical sources: number theory, the theory of algebraic equations, and geometry. The number-theoretic strand was begun by Leonhard Euler, and developed by Gauss's work on modular arithmetic and additive and multiplicative groups related to quadratic fields. Early results about permutation groups were obtained by Lagrange, Ruffini, and Abel in their quest for general solutions of polynomial equations of high degree. Évariste Galois coined the term “group” and established a connection, now known as Galois theory, between the nascent theory of groups and field theory. In geometry, groups first became important in projective geometry and, later, non-Euclidean geometry. Felix Klein's Erlangen program famously proclaimed group theory to be the organizing principle of geometry. Galois, in the 1830s, was the first to employ groups to determine the solvability of polynomial equations. Arthur Cayley and Augustin Louis Cauchy pushed these investigations further by creating the theory of permutation group. The second historical source for groups stems from geometrical situations. In an attempt to come to grips with possible geometries (such as euclidean, hyperbolic or projective geometry) using group theory, Felix Klein initiated the Erlangen programme. Sophus Lie, in 1884, started using groups (now called Lie groups) attached to analytic problems. Thirdly, groups were (first implicitly and later explicitly) used in algebraic number theory. The different scope of these early sources resulted in different notions of groups. The theory of groups was unified starting around 1880. Since then, the impact of group theory has been ever growing, giving rise to the birth of abstract algebra in the early 20th century, representation theory, and many more influential spin-off domains. The classification of finite simple groups is a vast body of work from the mid 20th century, classifying all the finite simple groups.

Main classes of groups The range of groups being considered has gradually expanded from finite permutation groups and special examples of matrix groups to abstract groups that may be specified through a presentation by generators and relations.

Permutation groups The first class of groups to undergo a systematic study was permutation groups. Given any set X and a collection G of bijections of X into itself (known as permutations) that is closed under compositions and inverses, G is a group acting on X. If X consists of n elements and G consists of all permutations, G is the symmetric group Sn; in general, G is a subgroup of the symmetric group of X. An early construction due to Cayley exhibited any group as a permutation group, acting on itself (X = G) by means of the left regular representation.

27

Group theory In many cases, the structure of a permutation group can be studied using the properties of its action on the corresponding set. For example, in this way one proves that for n ≥ 5, the alternating group An is simple, i.e. does not admit any proper normal subgroups. This fact plays a key role in the impossibility of solving a general algebraic equation of degree n ≥ 5 in radicals.

Matrix groups The next important class of groups is given by matrix groups, or linear groups. Here G is a set consisting of invertible matrices of given order n over a field K that is closed under the products and inverses. Such a group acts on the n-dimensional vector space Kn by linear transformations. This action makes matrix groups conceptually similar to permutation groups, and geometry of the action may be usefully exploited to establish properties of the group G.

Transformation groups Permutation groups and matrix groups are special cases of transformation groups: groups that act on a certain space X preserving its inherent structure. In the case of permutation groups, X is a set; for matrix groups, X is a vector space. The concept of a transformation group is closely related with the concept of a symmetry group: transformation groups frequently consist of all transformations that preserve a certain structure. The theory of transformation groups forms a bridge connecting group theory with differential geometry. A long line of research, originating with Lie and Klein, considers group actions on manifolds by homeomorphisms or diffeomorphisms. The groups themselves may be discrete or continuous.

Abstract groups Most groups considered in the first stage of the development of group theory were "concrete", having been realized through numbers, permutations, or matrices. It was not until the late nineteenth century that the idea of an abstract group as a set with operations satisfying a certain system of axioms began to take hold. A typical way of specifying an abstract group is through a presentation by generators and relations,

A significant source of abstract groups is given by the construction of a factor group, or quotient group, G/H, of a group G by a normal subgroup H. Class groups of algebraic number fields were among the earliest examples of factor groups, of much interest in number theory. If a group G is a permutation group on a set X, the factor group G/H is no longer acting on X; but the idea of an abstract group permits one not to worry about this discrepancy. The change of perspective from concrete to abstract groups makes it natural to consider properties of groups that are independent of a particular realization, or in modern language, invariant under isomorphism, as well as the classes of group with a given such property: finite groups, periodic groups, simple groups, solvable groups, and so on. Rather than exploring properties of an individual group, one seeks to establish results that apply to a whole class of groups. The new paradigm was of paramount importance for the development of mathematics: it foreshadowed the creation of abstract algebra in the works of Hilbert, Emil Artin, Emmy Noether, and mathematicians of their school.

28

Group theory

Topological and algebraic groups An important elaboration of the concept of a group occurs if G is endowed with additional structure, notably, of a topological space, differentiable manifold, or algebraic variety. If the group operations m (multiplication) and i (inversion),

are compatible with this structure, i.e. are continuous, smooth or regular (in the sense of algebraic geometry) maps then G becomes a topological group, a Lie group, or an algebraic group.[1] The presence of extra structure relates these types of groups with other mathematical disciplines and means that more tools are available in their study. Topological groups form a natural domain for abstract harmonic analysis, whereas Lie groups (frequently realized as transformation groups) are the mainstays of differential geometry and unitary representation theory. Certain classification questions that cannot be solved in general can be approached and resolved for special subclasses of groups. Thus, compact connected Lie groups have been completely classified. There is a fruitful relation between infinite abstract groups and topological groups: whenever a group Γ can be realized as a lattice in a topological group G, the geometry and analysis pertaining to G yield important results about Γ. A comparatively recent trend in the theory of finite groups exploits their connections with compact topological groups (profinite groups): for example, a single p-adic analytic group G has a family of quotients which are finite p-groups of various orders, and properties of G translate into the properties of its finite quotients.

Combinatorial and geometric group theory Groups can be described in different ways. Finite groups can be described by writing down the group table consisting of all possible multiplications g • h. A more important way of defining a group is by generators and relations, also called the presentation of a group. Given any set F of generators {gi}i ∈ I, the free group generated by F surjects onto the group G. The kernel of this map is called subgroup of relations, generated by some subset D. The presentation is usually denoted by 〈F | D 〉. For example, the group Z = 〈a | 〉 can be generated by one element a (equal to +1 or −1) and no relations, because n·1 never equals 0 unless n is zero. A string consisting of generator symbols is called a word. Combinatorial group theory studies groups from the perspective of generators and relations.[2] It is particularly useful where finiteness assumptions are satisfied, for example finitely generated groups, or finitely presented groups (i.e. in addition the relations are finite). The area makes use of the connection of graphs via their fundamental groups. For example, one can show that every subgroup of a free group is free. There are several natural questions arising from giving a group by its presentation. The word problem asks whether two words are effectively the same group element. By relating the problem to Turing machines, one can show that there is in general no algorithm solving this task. An equally difficult problem is, whether two groups given by different presentations are actually isomorphic. For example Z can also be presented by 〈x, y | xyxyx = 1〉 and it is not obvious (but true) that this presentation is isomorphic to the standard one above.

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Group theory

Geometric group theory attacks these problems from a geometric viewpoint, either by viewing groups as geometric objects, or by finding suitable geometric objects a group acts on.[3] The first idea is made precise by means of the Cayley graph, whose vertices correspond to group elements and edges correspond to right multiplication in the group. Given two elements, one constructs the word metric given by the length of the minimal path between the elements. A theorem of Milnor and Svarc then says that given a group G acting in a reasonable manner on a metric space X, for example a compact manifold, then G is quasi-isometric (i.e. looks similar from the far) to the space X.

30

The Cayley graph of 〈 x, y ∣ 〉, the free group of rank 2.

Representation of groups Saying that a group G acts on a set X means that every element defines a bijective map on a set in a way compatible with the group structure. When X has more structure, it is useful to restrict this notion further: a representation of G on a vector space V is a group homomorphism: ρ : G → GL(V), where GL(V) consists of the invertible linear transformations of V. In other words, to every group element g is assigned an automorphism ρ(g) such that ρ(g) ∘ ρ(h) = ρ(gh) for any h in G. This definition can be understood in two directions, both of which give rise to whole new domains of mathematics.[4] On the one hand, it may yield new information about the group G: often, the group operation in G is abstractly given, but via ρ, it corresponds to the multiplication of matrices, which is very explicit.[5] On the other hand, given a well-understood group acting on a complicated object, this simplifies the study of the object in question. For example, if G is finite, it is known that V above decomposes into irreducible parts. These parts in turn are much more easily manageable than the whole V (via Schur's lemma). Given a group G, representation theory then asks what representations of G exist. There are several settings, and the employed methods and obtained results are rather different in every case: representation theory of finite groups and representations of Lie groups are two main subdomains of the theory. The totality of representations is governed by the group's characters. For example, Fourier polynomials can be interpreted as the characters of U(1), the group of complex numbers of absolute value 1, acting on the L2-space of periodic functions.

Connection of groups and symmetry Given a structured object X of any sort, a symmetry is a mapping of the object onto itself which preserves the structure. This occurs in many cases, for example 1. If X is a set with no additional structure, a symmetry is a bijective map from the set to itself, giving rise to permutation groups. 2. If the object X is a set of points in the plane with its metric structure or any other metric space, a symmetry is a bijection of the set to itself which preserves the distance between each pair of points (an isometry). The corresponding group is called isometry group of X. 3. If instead angles are preserved, one speaks of conformal maps. Conformal maps give rise to Kleinian groups, for example. 4. Symmetries are not restricted to geometrical objects, but include algebraic objects as well. For instance, the equation

Group theory

has the two solutions

31

, and

. In this case, the group that exchanges the two roots is the Galois

group belonging to the equation. Every polynomial equation in one variable has a Galois group, that is a certain permutation group on its roots. The axioms of a group formalize the essential aspects of symmetry. Symmetries form a group: they are closed because if you take a symmetry of an object, and then apply another symmetry, the result will still be a symmetry. The identity keeping the object fixed is always a symmetry of an object. Existence of inverses is guaranteed by undoing the symmetry and the associativity comes from the fact that symmetries are functions on a space, and composition of functions are associative. Frucht's theorem says that every group is the symmetry group of some graph. So every abstract group is actually the symmetries of some explicit object. The saying of "preserving the structure" of an object can be made precise by working in a category. Maps preserving the structure are then the morphisms, and the symmetry group is the automorphism group of the object in question.

Applications of group theory Applications of group theory abound. Almost all structures in abstract algebra are special cases of groups. Rings, for example, can be viewed as abelian groups (corresponding to addition) together with a second operation (corresponding to multiplication). Therefore group theoretic arguments underlie large parts of the theory of those entities. Galois theory uses groups to describe the symmetries of the roots of a polynomial (or more precisely the automorphisms of the algebras generated by these roots). The fundamental theorem of Galois theory provides a link between algebraic field extensions and group theory. It gives an effective criterion for the solvability of polynomial equations in terms of the solvability of the corresponding Galois group. For example, S5, the symmetric group in 5 elements, is not solvable which implies that the general quintic equation cannot be solved by radicals in the way equations of lower degree can. The theory, being one of the historical roots of group theory, is still fruitfully applied to yield new results in areas such as class field theory. Algebraic topology is another domain which prominently associates groups to the objects the theory is interested in. There, groups are used to describe certain invariants of topological spaces. They are called "invariants" because they are defined in such a way that they do not change if the space is subjected to some deformation. For example, the fundamental group "counts" how many paths in the space are essentially different. The Poincaré conjecture, proved in 2002/2003 by Grigori Perelman is a prominent application of this idea. The influence is not unidirectional, though. For example, algebraic topology makes use of Eilenberg–MacLane spaces which are spaces with prescribed homotopy groups. Similarly algebraic K-theory stakes in a crucial way on classifying spaces of groups. Finally, the name of the torsion subgroup of an infinite group shows the legacy of topology in group theory. Algebraic geometry and cryptography likewise uses group theory in many ways. Abelian varieties have been introduced above. The presence of the group operation yields additional information which makes these varieties particularly accessible. They also often serve as a test for new conjectures.[6] The one-dimensional case, namely elliptic curves is studied in particular detail. They are both theoretically and practically intriguing.[7] Very large groups of prime order constructed in Elliptic-Curve Cryptography serve for public key cryptography. Cryptographical methods of this kind

A torus. Its abelian group structure is induced from the map C → C/Z+τZ, where τ is a parameter.

Group theory

The cyclic group Z26 underlies Caesar's cipher.

32 benefit from the flexibility of the geometric objects, hence their group structures, together with the complicated structure of these groups, which make the discrete logarithm very hard to calculate. One of the earliest encryption protocols, Caesar's cipher, may also be interpreted as a (very easy) group operation. In another direction, toric varieties are algebraic varieties acted on by a torus. Toroidal embeddings have recently led to advances in algebraic geometry, in particular resolution of singularities.[8]

Algebraic number theory is a special case of group theory, thereby following the rules of the latter. For example, Euler's product formula

captures the fact that any integer decomposes in a unique way into primes. The failure of this statement for more general rings gives rise to class groups and regular primes, which feature in Kummer's treatment of Fermat's Last Theorem. • The concept of the Lie group (named after mathematician Sophus Lie) is important in the study of differential equations and manifolds; they describe the symmetries of continuous geometric and analytical structures. Analysis on these and other groups is called harmonic analysis. Haar measures, that is integrals invariant under the translation in a Lie group, are used for pattern recognition and other image processing techniques.[9] • In combinatorics, the notion of permutation group and the concept of group action are often used to simplify the counting of a set of objects; see in particular Burnside's lemma. • The presence of the 12-periodicity in the circle of fifths yields applications of elementary group theory in musical set theory. • In physics, groups are important because they describe the symmetries which the laws of physics seem to obey. Physicists are very interested in group representations, especially of Lie groups, since these representations often point the way to the "possible" physical theories. Examples of the use of groups in physics include the Standard Model, gauge theory, the Lorentz group, and the Poincaré group. • In chemistry and materials science, groups are used to classify crystal structures, regular polyhedra, and the symmetries of molecules. The assigned point groups can then be used to determine physical properties (such as polarity and chirality), spectroscopic properties (particularly useful for Raman spectroscopy and infrared spectroscopy), and to construct molecular orbitals.

The circle of fifths may be endowed with a cyclic group structure

Group theory

See also • Group (mathematics) • Glossary of group theory • List of group theory topics

Notes [1] This process of imposing extra structure has been formalized through the notion of a group object in a suitable category. Thus Lie groups are group objects in the category of differentiable manifolds and affine algebraic groups are group objects in the category of affine algebraic varieties. [2] Schupp & Lyndon 2001 [3] La Harpe 2000 [4] Such as group cohomology or equivariant K-theory. [5] In particular, if the representation is faithful. [6] For example the Hodge conjecture (in certain cases). [7] See the Birch-Swinnerton-Dyer conjecture, one of the millennium problems [8] Abramovich, Dan; Karu, Kalle; Matsuki, Kenji; Wlodarczyk, Jaroslaw (2002), "Torification and factorization of birational maps", Journal of the American Mathematical Society 15 (3): 531–572, doi:10.1090/S0894-0347-02-00396-X, MR1896232 [9] Lenz, Reiner (1990), Group theoretical methods in image processing (http:/ / webstaff. itn. liu. se/ ~reile/ LNCS413/ index. htm), Lecture Notes in Computer Science, 413, Berlin, New York: Springer-Verlag, doi:10.1007/3-540-52290-5, ISBN 978-0-387-52290-6,

References • Borel, Armand (1991), Linear algebraic groups, Graduate Texts in Mathematics, 126 (2nd ed.), Berlin, New York: Springer-Verlag, MR1102012, ISBN 978-0-387-97370-8 • Carter, Nathan C. (2009), Visual group theory (http://web.bentley.edu/empl/c/ncarter/vgt/), Classroom Resource Materials Series, Mathematical Association of America, MR2504193, ISBN 978-0-88385-757-1 • Cannon, John J. (1969), "Computers in group theory: A survey", Communications of the Association for Computing Machinery 12: 3–12, doi:10.1145/362835.362837, MR0290613 • Frucht, R. (1939), "Herstellung von Graphen mit vorgegebener abstrakter Gruppe" (http://www.numdam.org/ numdam-bin/fitem?id=CM_1939__6__239_0), Compositio Mathematica 6: 239–50, ISSN 0010-437X • Golubitsky, Martin; Stewart, Ian (2006), "Nonlinear dynamics of networks: the groupoid formalism", Bull. Amer. Math. Soc. (N.S.) 43: 305–364, doi:10.1090/S0273-0979-06-01108-6, MR2223010 Shows the advantage of generalising from group to groupoid. • Judson, Thomas W. (1997), Abstract Algebra: Theory and Applications (http://abstract.ups.edu) An introductory undergraduate text in the spirit of texts by Gallian or Herstein, covering groups, rings, integral domains, fields and Galois theory. Free downloadable PDF with open-source GFDL license. • Kleiner, Israel (1986), "The evolution of group theory: a brief survey" (http://jstor.org/stable/2690312), Mathematics Magazine 59 (4): 195–215, doi:10.2307/2690312, MR863090, ISSN 0025-570X • La Harpe, Pierre de (2000), Topics in geometric group theory, University of Chicago Press, ISBN 978-0-226-31721-2 • Livio, M. (2005), The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry, Simon & Schuster, ISBN 0-7432-5820-7 Conveys the practical value of group theory by explaining how it points to symmetries in physics and other sciences. • Mumford, David (1970), Abelian varieties, Oxford University Press, ISBN 978-0-19-560528-0, OCLC 138290 • Ronan M., 2006. Symmetry and the Monster. Oxford University Press. ISBN 0-19-280722-6. For lay readers. Describes the quest to find the basic building blocks for finite groups. • Rotman, Joseph (1994), An introduction to the theory of groups, New York: Springer-Verlag, ISBN 0-387-94285-8 A standard contemporary reference.

33

Group theory

34

• Schupp, Paul E.; Lyndon, Roger C. (2001), Combinatorial group theory, Berlin, New York: Springer-Verlag, ISBN 978-3-540-41158-1 • Scott, W. R. (1987) [1964], Group Theory, New York: Dover, ISBN 0-486-65377-3 Inexpensive and fairly readable, but somewhat dated in emphasis, style, and notation. • Shatz, Stephen S. (1972), Profinite groups, arithmetic, and geometry, Princeton University Press, MR0347778, ISBN 978-0-691-08017-8 • Weibel, Charles A. (1994), An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, 38, Cambridge University Press, MR1269324, ISBN 978-0-521-55987-4, OCLC 36131259

External links • History of the abstract group concept (http://www-history.mcs.st-andrews.ac.uk/history/HistTopics/ Abstract_groups.html) • Higher dimensional group theory (http://www.bangor.ac.uk/r.brown/hdaweb2.htm) This presents a view of group theory as level one of a theory which extends in all dimensions, and has applications in homotopy theory and to higher dimensional nonabelian methods for local-to-global problems. • Plus teacher and student package: Group Theory (http://plus.maths.org/issue48/package/index.html) This package brings together all the articles on group theory from Plus, the online mathematics magazine produced by the Millennium Mathematics Project at the University of Cambridge, exploring applications and recent breakthroughs, and giving explicit definitions and examples of groups. • US Naval Academy group theory guide (http://www.usna.edu/Users/math/wdj/tonybook/gpthry/node1. html) A general introduction to group theory with exercises written by Tony Gaglione.

Elementary group theory In mathematics and abstract algebra, a group is the algebraic structure denotes a binary operation

be arbitrary elements of

is normally

. Then:

. This axiom is often omitted because a binary operation is closed by definition.

• A2, Associativity. . • A3, Identity. There exists an identity (or neutral) element of is unique by Theorem 1.4 below. • A4, Inverse. For each , there exists an inverse element inverse of

is a non-empty set and

called the group operation. The notation

shortened to the infix notation , or even to . A group must obey the following rules (or axioms). Let • A1, Closure.

, where

is unique by Theorem 1.5 below.

An abelian group also obeys the additional rule: • A5, Commutativity.

.

such that such that

. The identity . The

Elementary group theory

35

Notation The group

is often referred to as "the group

" or more simply as "

" is fundamental to the description of the group. we wish to assert that The group operation

" Nevertheless, the operation "

is usually read as "the group

under

". When

is a group (for example, when stating a theorem), we say that " is a group under ". can be interpreted in a great many ways. The generic notation for the group operation,

identity element, and inverse of

are

respectively. Because the group operation associates, parentheses

have only one necessary use in group theory: to set the scope of the inverse operation. Group theory may also be notated: • Additively by replacing the generic notation by

, with "+" being infix. Additive notation is typically

used when numerical addition or a commutative operation other than multiplication interprets the group operation; • Multiplicatively by replacing the generic notation by . Infix "*" is often replaced by simple concatenation, as in standard algebra. Multiplicative notation is typically used when numerical multiplication or a noncommutative operation interprets the group operation. Other notations are of course possible.

Examples Arithmetic • Take • Take

or

or or or

, then or

is an abelian group. , then is an abelian group.

Function composition • Let

be an arbitrary set, and let

composition, notated by infix is

be the set of all bijective functions from

, interpret the group operation. Then

to

. Let function

is a group whose identity element

The group inverse of an arbitrary group element

is the function inverse

Alternative Axioms The pair of axioms A3 and A4 may be replaced either by the pair: • A3’, left neutral. There exists an • A4’, left inverse. For each

such that for all , there exists an element

,

. such that

.

or by the pair: • A3”, right neutral. There exists an • A4”, right inverse. For each

such that for all , there exists an element

,

. such that

.

These evidently weaker axiom pairs are trivial consequences of A3 and A4. We will now show that the nontrivial converse is also true. Given a left neutral element and for any given then A4’ says there exists an such that

.

Theorem 1.2: Proof. Let

be an inverse of

Then:

Elementary group theory

36

This establishes A4 (and hence A4”). Theorem 1.2a: Proof.

This establishes A3 (and hence A3”). Theorem: Given A1 and A2, A3’ and A4’ imply A3 and A4. Proof. Theorems 1.2 and 1.2a. Theorem: Given A1 and A2, A3” and A4” imply A3 and A4. Proof. Similar to the above.

Basic theorems Identity is unique Theorem 1.4: The identity element of a group Proof: Suppose that

and

is unique.

are two identity elements of

. Then

As a result, we can speak of the identity element of groups are being discussed and compared,

rather than an identity element. Where different

denotes the identity of the specific group

.

Inverses are unique Theorem 1.5: The inverse of each element in Proof: Suppose that

and

are two inverses of an element

As a result, we can speak of the inverse of an element , we denote by

is unique.

the unique inverse of

.

of

. Then

, rather than an inverse. Without ambiguity, for all

in

Elementary group theory

37

Inverting twice takes you back to where you started Theorem 1.6: For all elements Proof.

in a group

and

.

are both true by A4. Therefore both

and

are inverses of

By

Theorem 1.5,

Inverse of ab Theorem 1.7: For all elements

and

in group

,

.

Proof.

. The conclusion follows

from Theorem 1.4.

Cancellation Theorem

1.8:

For

all

elements

in

a

group

,

then

. Proof. (1) If

, then multiplying by the same value on either side preserves equality.

(2) If

then by (1)

(3) If

we use the same method as in (2).

Latin square property Theorem 1.3: For all elements namely Proof. Existence: If we let Unicity: Suppose

in a group

, there exists a unique

and

,

. , then satisfies

. , then by Theorem 1.8,

Powers For

such that

in group

Theorem 1.9: For all

in group

we define:

and

:

.

Elementary group theory

Order Of a group element The order of an element a in a group G is the least positive integer n such that an = e. Sometimes this is written "o(a)=n". n can be infinite. Theorem 1.10: A group whose nontrivial elements all have order 2 is abelian. In other words, if all elements g in a group G g*g=e is the case, then for all elements a,b in G, a*b=b*a. Proof. Let a, b be any 2 elements in the group G. By A1, a*b is also a member of G. Using the given condition, we know that (a*b)*(a*b)=e. Hence: • • • • • •

b*a =e*(b*a)*e = (a*a)*(b*a)*(b*b) =a*(a*b)*(a*b)*b =a*e*b =a*b.

Since the group operation * commutes, the group is abelian

Of a group The order of the group G, usually denoted by |G| or occasionally by o(G), is the number of elements in the set G, in which case is a finite group. If G is an infinite set, then the group has order equal to the cardinality of G, and is an infinite group.

Subgroups A subset H of G is called a subgroup of a group if H satisfies the axioms of a group, using the same operator "*", and restricted to the subset H. Thus if H is a subgroup of , then is also a group, and obeys the above theorems, restricted to H. The order of subgroup H is the number of elements in H. A proper subgroup of a group G is a subgroup which is not identical to G. A non-trivial subgroup of G is (usually) any proper subgroup of G which contains an element other than e. Theorem 2.1: If H is a subgroup of , then the identity eH in H is identical to the identity e in (G,*). Proof. If h is in H, then h*eH = h; since h must also be in G, h*e = h; so by theorem 1.8, eH = e. Theorem 2.2: If H is a subgroup of G, and h is an element of H, then the inverse of h in H is identical to the inverse of h in G. Proof. Let h and k be elements of H, such that h*k = e; since h must also be in G, h*h -1 = e; so by theorem 1.5, k = h -1. Given a subset S of G, we often want to determine whether or not S is also a subgroup of G. A handy theorem valid for both infinite and finite groups is: Theorem 2.3: If S is a non-empty subset of G, then S is a subgroup of G if and only if for all a,b in S, a*b -1 is in S. Proof. If for all a, b in S, a*b -1 is in S, then • e is in S, since a*a -1 = e is in S. • for all a in S, e*a -1 = a -1 is in S • for all a, b in S, a*b = a*(b -1) -1 is in S Thus, the axioms of closure, identity, and inverses are satisfied, and associativity is inherited; so S is subgroup. Conversely, if S is a subgroup of G, then it obeys the axioms of a group.

38

Elementary group theory • As noted above, the identity in S is identical to the identity e in G. • By A4, for all b in S, b -1 is in S • By A1, a*b -1 is in S. The intersection of two or more subgroups is again a subgroup. Theorem 2.4: The intersection of any non-empty set of subgroups of a group G is a subgroup. Proof. Let {Hi} be a set of subgroups of G, and let K = ∩{Hi}. e is a member of every Hi by theorem 2.1; so K is not empty. If h and k are elements of K, then for all i, • h and k are in Hi. • By the previous theorem, h*k -1 is in Hi • Therefore, h*k -1 is in ∩{Hi}. Therefore for all h, k in K, h*k -1 is in K. Then by the previous theorem, K=∩{Hi} is a subgroup of G; and in fact K is a subgroup of each Hi. Given a group , define x*x as x², x*x*x*...*x (n times) as xn, and define x0 = e. Similarly, let x -n for (x -1)n. Then we have: Theorem 2.5: Let a be an element of a group (G,*). Then the set {an: n is an integer} is a subgroup of G. A subgroup of this type is called a cyclic subgroup; the subgroup of the powers of a is often written as , and we say that a generates .

Cosets If S and T are subsets of G, and a is an element of G, we write "a*S" to refer to the subset of G made up of all elements of the form a*s, where s is an element of S; similarly, we write "S*a" to indicate the set of elements of the form s*a. We write S*T for the subset of G made up of elements of the form s*t, where s is an element of S and t is an element of T. If H is a subgroup of G, then a left coset of H is a set of the form a*H, for some a in G. A right coset is a subset of the form H*a. If H is a subgroup of G, the following useful theorems, stated without proof, hold for all cosets: • And x and y are elements of G, then either x*H = y*H, or x*H and y*H have empty intersection. • Every left (right) coset of H in G contains the same number of elements. • G is the disjoint union of the left (right) cosets of H. • Then the number of distinct left cosets of H equals the number of distinct right cosets of H. Define the index of a subgroup H of a group G (written "[G:H]") to be the number of distinct left cosets of H in G. From these theorems, we can deduce the important Lagrange's theorem, relating the order of a subgroup to the order of a group: • Lagrange's theorem: If H is a subgroup of G, then |G| = |H|*[G:H]. For finite groups, this can be restated as: • Lagrange's theorem: If H is a subgroup of a finite group G, then the order of H divides the order of G. • If the order of group G is a prime number, G is cyclic.

39

Elementary group theory

References • Jordan, C. R and D.A. Groups. Newnes (Elsevier), ISBN 0-340-61045-X • Scott, W R. Group Theory. Dover Publications, ISBN 0-486-65377-3

Symmetry group The symmetry group of an object (image, signal, etc.) is the group of all isometries under which it is invariant with composition as the operation. It is a subgroup of the isometry group of the space concerned. If not stated otherwise, this article considers symmetry groups in Euclidean geometry, but the concept may also be studied in wider contexts; see below.

Introduction The "objects" may be geometric figures, images, and patterns, such as a wallpaper pattern. The definition can be made more precise by specifying what is meant by image or pattern, e.g., a function of position with values in a set of colors. For symmetry of physical objects, one may also want to take physical composition into account. The group of isometries of space induces a group action on objects in it. The symmetry group is sometimes also called full A tetrahedron can be placed in 12 distinct positions by rotation alone. symmetry group in order to emphasize that it These are illustrated above in the cycle graph format, along with the 180° includes the orientation-reversing isometries (like edge (blue arrows) and 120° vertex (reddish arrows) rotations that permute reflections, glide reflections and improper the tetrahedron through the positions. The 12 rotations form the rotation rotations) under which the figure is invariant. The (symmetry) group of the figure. subgroup of orientation-preserving isometries (i.e. translations, rotations, and compositions of these) which leave the figure invariant is called its proper symmetry group. The proper symmetry group of an object is equal to its full symmetry group if and only if the object is chiral (and thus there are no orientation-reversing isometries under which it is invariant). Any symmetry group whose elements have a common fixed point, which is true for all finite symmetry groups and also for the symmetry groups of bounded figures, can be represented as a subgroup of orthogonal group O(n) by choosing the origin to be a fixed point. The proper symmetry group is a subgroup of the special orthogonal group SO(n) then, and therefore also called rotation group of the figure. Discrete symmetry groups come in three types: (1) finite point groups, which include only rotations, reflections, inversion and rotoinversion - they are in fact just the finite subgroups of O(n), (2) infinite lattice groups, which include only translations, and (3) infinite space groups which combines elements of both previous types, and may also include extra transformations like screw axis and glide reflection. There are also continuous symmetry groups,

40

Symmetry group which contain rotations of arbitrarily small angles or translations of arbitrarily small distances. The group of all symmetries of a sphere O(3) is an example of this, and in general such continuous symmetry groups are studied as Lie groups. With a categorization of subgroups of the Euclidean group corresponds a categorization of symmetry groups. Two geometric figures are considered to be of the same symmetry type if their symmetry groups are conjugate subgroups of the Euclidean group E(n) (the isometry group of Rn), where two subgroups H1, H2 of a group G are conjugate, if there exists g ∈ G such that H1=g−1H2g. For example: • two 3D figures have mirror symmetry, but with respect to different mirror planes. • two 3D figures have 3-fold rotational symmetry, but with respect to different axes. • two 2D patterns have translational symmetry, each in one direction; the two translation vectors have the same length but a different direction. When considering isometry groups, one may restrict oneself to those where for all points the set of images under the isometries is topologically closed. This excludes for example in 1D the group of translations by a rational number. A "figure" with this symmetry group is non-drawable and up to arbitrarily fine detail homogeneous, without being really homogeneous.

One dimension The isometry groups in 1D where for all points the set of images under the isometries is topologically closed are: • • • •

the trivial group C1 the groups of two elements generated by a reflection in a point; they are isomorphic with C2 the infinite discrete groups generated by a translation; they are isomorphic with Z the infinite discrete groups generated by a translation and a reflection in a point; they are isomorphic with the generalized dihedral group of Z, Dih(Z), also denoted by D∞ (which is a semidirect product of Z and C2). • the group generated by all translations (isomorphic with R); this group cannot be the symmetry group of a "pattern": it would be homogeneous, hence could also be reflected. However, a uniform 1D vector field has this symmetry group. • the group generated by all translations and reflections in points; they are isomorphic with the generalized dihedral group of R, Dih(R). See also symmetry groups in one dimension.

Two dimensions Up to conjugacy the discrete point groups in 2 dimensional space are the following classes: • cyclic groups C1, C2, C3, C4,... where Cn consists of all rotations about a fixed point by multiples of the angle 360°/n • dihedral groups D1, D2, D3, D4,... where Dn (of order 2n) consists of the rotations in Cn together with reflections in n axes that pass through the fixed point. C1 is the trivial group containing only the identity operation, which occurs when the figure has no symmetry at all, for example the letter F. C2 is the symmetry group of the letter Z, C3 that of a triskelion, C4 of a swastika, and C5, C6 etc. are the symmetry groups of similar swastika-like figures with five, six etc. arms instead of four. D1 is the 2-element group containing the identity operation and a single reflection, which occurs when the figure has only a single axis of bilateral symmetry, for example the letter A. D2, which is isomorphic to the Klein four-group, is the symmetry group of a non-equilateral rectangle, and D3, D4 etc. are the symmetry groups of the regular polygons. The actual symmetry groups in each of these cases have two degrees of freedom for the center of rotation, and in the case of the dihedral groups, one more for the positions of the mirrors.

41

Symmetry group

42

The remaining isometry groups in 2D with a fixed point, where for all points the set of images under the isometries is topologically closed are: • the special orthogonal group SO(2) consisting of all rotations about a fixed point; it is also called the circle group S1, the multiplicative group of complex numbers of absolute value 1. It is the proper symmetry group of a circle and the continuous equivalent of Cn. There is no figure which has as full symmetry group the circle group, but for a vector field it may apply (see the 3D case below). • the orthogonal group O(2) consisting of all rotations about a fixed point and reflections in any axis through that fixed point. This is the symmetry group of a circle. It is also called Dih(S1) as it is the generalized dihedral group of S1. For non-bounded figures, the additional isometry groups can include translations; the closed ones are: • the 7 frieze groups • the 17 wallpaper groups • for each of the symmetry groups in 1D, the combination of all symmetries in that group in one direction, and the group of all translations in the perpendicular direction • ditto with also reflections in a line in the first direction

Three dimensions Up to conjugacy the set of 3D point groups consists of 7 infinite series, and 7 separate ones. In crystallography they are restricted to be compatible with the discrete translation symmetries of a crystal lattice. This crystallographic restriction of the infinite families of general point groups results in 32 crystallographic point groups (27 from the 7 infinite series, and 5 of the 7 others). The continuous symmetry groups with a fixed point include those of: • cylindrical symmetry without a symmetry plane perpendicular to the axis, this applies for example often for a bottle • cylindrical symmetry with a symmetry plane perpendicular to the axis • spherical symmetry For objects and scalar fields the cylindrical symmetry implies vertical planes of reflection. However, for vector fields it does not: in cylindrical coordinates with respect to some axis, has cylindrical symmetry with respect to the axis if and only if

and

have this symmetry, i.e., they do not depend on φ.

Additionally there is reflectional symmetry if and only if . For spherical symmetry there is no such distinction, it implies planes of reflection. The continuous symmetry groups without a fixed point include those with a screw axis, such as an infinite helix. See also subgroups of the Euclidean group.

Symmetry groups in general In wider contexts, a symmetry group may be any kind of transformation group, or automorphism group. Once we know what kind of mathematical structure we are concerned with, we should be able to pinpoint what mappings preserve the structure. Conversely, specifying the symmetry can define the structure, or at least clarify what we mean by an invariant, geometric language in which to discuss it; this is one way of looking at the Erlangen programme. For example, automorphism groups of certain models of finite geometries are not "symmetry groups" in the usual sense, although they preserve symmetry. They do this by preserving families of point-sets rather than point-sets (or "objects") themselves. Like above, the group of automorphisms of space induces a group action on objects in it.

Symmetry group For a given geometric figure in a given geometric space, consider the following equivalence relation: two automorphisms of space are equivalent if and only if the two images of the figure are the same (here "the same" does not mean something like e.g. "the same up to translation and rotation", but it means "exactly the same"). Then the equivalence class of the identity is the symmetry group of the figure, and every equivalence class corresponds to one isomorphic version of the figure. There is a bijection between every pair of equivalence classes: the inverse of a representative of the first equivalence class, composed with a representative of the second. In the case of a finite automorphism group of the whole space, its order is the order of the symmetry group of the figure multiplied by the number of isomorphic versions of the figure. Examples: • Isometries of the Euclidean plane, the figure is a rectangle: there are infinitely many equivalence classes; each contains 4 isometries. • The space is a cube with Euclidean metric; the figures include cubes of the same size as the space, with colors or patterns on the faces; the automorphisms of the space are the 48 isometries; the figure is a cube of which one face has a different color; the figure has a symmetry group of 8 isometries, there are 6 equivalence classes of 8 isometries, for 6 isomorphic versions of the figure. Compare Lagrange's theorem (group theory) and its proof.

Further reading • Burns, G.; Glazer, A.M. (1990). Space Groups for Scientists and Engineers (2nd ed.). Boston: Academic Press, Inc. ISBN 0-12-145761-3. • Clegg, W (1998). Crystal Structure Determination (Oxford Chemistry Primer). Oxford: Oxford University Press. ISBN 0-19-855-901-1. • O'Keeffe, M.; Hyde, B.G. (1996). Crystal Structures; I. Patterns and Symmetry. Washington, DC: Mineralogical Society of America, Monograph Series. ISBN 0-939950-40-5. • Miller, Willard Jr. (1972). Symmetry Groups and Their Applications [1]. New York: Academic Press. OCLC 589081. Retrieved 2009-09-28.

External links • Weisstein, Eric W., "Symmetry Group [2]" from MathWorld. • Weisstein, Eric W., "Tetrahedral Group [3]" from MathWorld. • Overview of the 32 crystallographic point groups [4] - form the first parts (apart from skipping n=5) of the 7 infinite series and 5 of the 7 separate 3D point groups

References [1] [2] [3] [4]

http:/ / www. ima. umn. edu/ ~miller/ symmetrygroups. html http:/ / mathworld. wolfram. com/ SymmetryGroup. html http:/ / mathworld. wolfram. com/ TetrahedralGroup. html http:/ / newton. ex. ac. uk/ research/ qsystems/ people/ goss/ symmetry/ Solids. html

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Symmetric group

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Symmetric group In mathematics, the symmetric group on a set is the group consisting of all bijections of the set (all one-to-one and onto functions) from the set to itself with function composition as the group operation.[1] The symmetric group is important to diverse areas of mathematics such as Galois theory, invariant theory, the representation theory of Lie groups, and combinatorics. Cayley's theorem states that every group G is isomorphic to a subgroup of the symmetric group on G. This article focuses on the finite symmetric groups: their applications, their elements, their conjugacy classes, a finite presentation, their subgroups, their automorphism groups, and their representation theory. For the remainder of this article, "symmetric group" will mean a symmetric group on a finite set.

Cayley graph of the symmetric group of degree 4 (S4) represented as the group of rotations of a standard die.

Definition and first properties The symmetric group on a set X is the group whose underlying set is the collection of all bijections from X to X and whose group operation is that of function composition.[1] The symmetric group of degree n is the symmetric group on the set X = { 1, 2, ..., n }. The symmetric group on a set X is denoted in various ways including SX, 1, 2, ..., n }, then the symmetric group on X is also denoted Sn,[1]

, ΣX, and Sym(X).[1] If X is the set {

Σn, and Sym(n).

Symmetric groups on infinite sets behave quite differently than symmetric groups on finite sets, and are discussed in (Scott 1987, Ch. 11), (Dixon & Mortimer 1996, Ch. 8), and (Cameron 1999). This article concentrates on the finite symmetric groups. The symmetric group on a set of n has order n!.[2] It is abelian if and only if n ≤ 2. For empty set and the singleton set) the symmetric group is trivial (note that this agrees with

and

(the ), and in

these cases the alternating group equals the symmetric group, rather than being an index two subgroup. The group Sn is solvable if and only if n ≤ 4. This is an essential part of the proof of the Abel–Ruffini theorem that shows that for every n > 4 there are polynomials of degree n which are not solvable by radicals, i.e., the solutions cannot be expressed by performing a finite number of operations of addition, subtraction, multiplication, division and root extraction on the polynomial's coefficients.

Applications The symmetric group on a set of size n is the Galois group of the general polynomial of degree n and plays an important role in Galois theory. In invariant theory, the symmetric group acts on the variables of a multi-variate function, and the functions left invariant are the so-called symmetric functions. In the representation theory of Lie groups, the representation theory of the symmetric group plays a fundamental role through the ideas of Schur functors. In the theory of Coxeter groups, the symmetric group is the Coxeter group of type An and occurs as the Weyl group of the general linear group. In combinatorics, the symmetric groups, their elements (permutations), and their representations provide a rich source of problems involving Young tableaux, plactic monoids, and the Bruhat order. Subgroups of symmetric groups are called permutation groups and are widely studied because of their importance in understanding group actions, homogenous spaces, and automorphism groups of graphs, such as the

Symmetric group

45

Higman–Sims group and the Higman–Sims graph.

Elements The elements of the symmetric group on a set X are the permutations of X.

Multiplication The group operation in a symmetric group is function composition, denoted by the symbol or simply by juxtaposition of the permutations. The composition of permutations f and g, pronounced "f after g", maps any element x of X to f(g(x)). Concretely, let

and (See permutation for an explanation of notation). Applying f after g maps 1 first to 2 and then 2 to itself; 2 to 5 and then to 4; 3 to 4 and then to 5, and so on. So composing f and g gives

A cycle of length L =k·m, taken to the k-th power, will decompose into k cycles of length m: For example (k=2, m=3),

Verification of group axioms To check that the symmetric group on a set X is indeed a group, it is necessary to verify the group axioms of associativity, identity, and inverses. The operation of function composition is always associative. The trivial bijection that assigns each element of X to itself serves as an identity for the group. Every bijection has an inverse function that undoes its action, and thus each element of a symmetric group does have an inverse.

Transpositions A transposition is a permutation which exchanges two elements and keeps all others fixed; for example (1 3) is a transposition. Every permutation can be written as a product of transpositions; for instance, the permutation g from above can be written as g = (1 5)(1 2)(3 4). Since g can be written as a product of an odd number of transpositions, it is then called an odd permutation, whereas f is an even permutation. The representation of a permutation as a product of transpositions is not unique; however, the number of transpositions needed to represent a given permutation is either always even or always odd. There are several short proofs of the invariance of this parity of a permutation. The product of two even permutations is even, the product of two odd permutations is even, and all other products are odd. Thus we can define the sign of a permutation:

With this definition,

Symmetric group

46

is a group homomorphism ({+1, –1} is a group under multiplication, where +1 is e, the neutral element). The kernel of this homomorphism, i.e. the set of all even permutations, is called the alternating group An. It is a normal subgroup of Sn, and for n ≥ 2 it has n! / 2 elements. The group Sn is the semidirect product of An and any subgroup generated by a single transposition. Furthermore, every permutation can be written as a product of adjacent transpositions, that is, transpositions of the form . For instance, the permutation g from above can also be written as g = (4 5)(3 4)(4 5)(1 2)(2 3)(3 4)(4 5). The representation of a permutation as a product of adjacent transpositions is also not unique.

Cycles A cycle of length k is a permutation f for which there exists an element x in {1,...,n} such that x, f(x), f2(x), ..., fk(x) = x are the only elements moved by f; it is required that k ≥ 2 since with k = 1 the element x itself would not be moved either. The permutation h defined by

is a cycle of length three, since h(1) = 4, h(4) = 3 and h(3) = 1, leaving 2 and 5 untouched. We denote such a cycle by (1 4 3). The order of a cycle is equal to its length. Cycles of length two are transpositions. Two cycles are disjoint if they move disjoint subsets of elements. Disjoint cycles commute, e.g. in S6 we have (3 1 4)(2 5 6) = (2 5 6)(3 1 4). Every element of Sn can be written as a product of disjoint cycles; this representation is unique up to the order of the factors.

Special elements Certain elements of the symmetric group of {1,2, ..., n} are of particular interest (these can be generalized to the symmetric group of any finite totally ordered set, but not to that of an unordered set). The order reversing permutation is the one given by:

This is the unique maximal element with respect to the Bruhat order and the longest element in the symmetric group with respect to generating set consisting of the adjacent transpositions (i i+1), 1 ≤ i ≤ n−1. This is an involution, and consists of ,

(non-adjacent) transpositions or

adjacent

transpositions:

so it thus has sign:

which is 4-periodic in n. In

, the perfect shuffle is the permutation that splits the set into 2 piles and interleaves them. Its sign is also

Note that the reverse on n elements and perfect shuffle on 2n elements have the same sign; these are important to the classification of Clifford algebras, which are 8-periodic.

Symmetric group

Conjugacy classes The conjugacy classes of Sn correspond to the cycle structures of permutations; that is, two elements of Sn are conjugate in Sn if and only if they consist of the same number of disjoint cycles of the same lengths. For instance, in S5, (1 2 3)(4 5) and (1 4 3)(2 5) are conjugate; (1 2 3)(4 5) and (1 2)(4 5) are not. A conjugating element of Sn can be constructed in "two line notation" by placing the "cycle notations" of the two conjugate permutations on top of one another. Continuing the previous example:

which can be written as the product of cycles, namely:

This permutation then relates (1 2 3)(4 5) and (1 4 3)(2 5) via conjugation, i.e.

It is clear that such a permutation is not unique.

Low degree groups The low-degree symmetric groups have simpler structure and exceptional structure and often must be treated separately. Sym(0) and Sym(1) The symmetric groups on the empty set and the singleton set are trivial, which corresponds to In this case the alternating group agrees with the symmetric group, rather than being an index 2 subgroup, and the sign map is trivial. Sym(2) The symmetric group on two points consists of exactly two elements: the identity and the permutation swapping the two points. It is a cyclic group and so abelian. In Galois theory, this corresponds to the fact that the quadratic formula gives a direct solution to the general quadratic polynomial after extracting only a single root. In invariant theory, the representation theory of the symmetric group on two points is quite simple and is seen as writing a function of two variables as a sum of its symmetric and anti-symmetric parts: Setting fs(x,y) = f(x,y) + f(y,x), and fa(x,y) = f(x,y) − f(y,x), one gets that 2·f = fs + fa. This process is known as symmetrization. Sym(3) is isomorphic to the dihedral group of order 6, the group of reflection and rotation symmetries of an equilateral triangle, since these symmetries permute the three vertices of the triangle. Cycles of length two correspond to reflections, and cycles of length three are rotations. In Galois theory, the sign map from Sym(3) to Sym(2) corresponds to the resolving quadratic for a cubic polynomial, as discovered by Gerolamo Cardano, while the Alt(3) kernel corresponds to the use of the discrete Fourier transform of order 3 in the solution, in the form of Lagrange resolvents. Sym(4) The group S4 is isomorphic to proper rotations of the cube; the isomorphism from the cube group to Sym(4) is given by the permutation action on the four diagonals of the cube. The group Alt(4) has a Klein four-group V as a proper normal subgroup, namely the double transpositions {(12)(34), (13)(24), (14)(23)}. This is also normal in Sym(4) with quotient Sym(3). In Galois theory, this map corresponds to the resolving cubic to a quartic polynomial, which allows the quartic to be solved by radicals, as established by Lodovico Ferrari. The Klein group can be understood in terms of the Lagrange resolvents of the quartic. The map from Sym(4) to Sym(3) also yields a 2-dimensional irreducible representation, which is an irreducible representation of a

47

Symmetric group

48

symmetric group of degree n of dimension below n−1, which only occurs for n=4. Sym(5) Sym(5) is the first non-solvable symmetric group. Along with the special linear group SL(2,5) and the icosahedral group Alt(5) × Sym(2), Sym(5) is one of the three non-solvable groups of order 120 up to isomorphism. Sym(5) is the Galois group of the general quintic equation, and the fact that Sym(5) is not a solvable group translates into the non-existence of a general formula to solve quintic polynomials by radicals. There is an exotic inclusion map as a transitive subgroup; the obvious inclusion map fixes a point and thus is not transitive. This yields the outer automorphism of

discussed below, and

corresponds to the resolvent sextic of a quintic. Sym(6) Sym(6), unlike other symmetric groups, has an outer automorphism. Using the language of Galois theory, this can also be understood in terms of Lagrange resolvents. The resolvent of a quintic is of degree 6—this corresponds to an exotic inclusion map as a transitive subgroup (the obvious inclusion map fixes a point and thus is not transitive) and, while this map does not make the general quintic solvable, it yields the exotic outer automorphism of

—see automorphisms of the symmetric and alternating

groups for details. Note that while Alt(6) and Alt(7) have an exceptional Schur multiplier (a triple cover) and that these extend to triple covers of Sym(6) and Sym(7), these do not correspond to exceptional Schur multipliers of the symmetric group.

Maps between symmetric groups Other than the trivial map

and the sign map

the notable maps between

symmetric groups, in order of relative dimension, are: •

corresponding to the exceptional normal subgroup

•

(or rather, a class of such maps up to inner automorphism) corresponding to the outer automorphism of

•

as a transitive subgroup, yielding the outer automorphism of

as discussed above.

Properties Symmetric groups are Coxeter groups and reflection groups. They can be realized as a group of reflections with respect to hyperplanes . Braid groups Bn admit symmetric groups Sn as quotient groups. Cayley's theorem states that every group G is isomorphic to a subgroup of the symmetric group on the elements of G, as a group acts on itself faithfully by (left or right) multiplication.

Relation with alternating group For n≥5, the alternating group

is simple, and the induced quotient is the sign map:

split by taking a transposition of two elements. Thus proper normal subgroups, as they would intersect

is the semidirect product

which is and has no other

in either the identity (and thus themselves be the identity or a

2-element group, which is not normal), or in (and thus themselves be or ). acts on its subgroup by conjugation, and for is the full automorphism group of Conjugation by even elements are inner automorphisms of of

of order 2 corresponds to conjugation by an odd element. For

automorphism of

so

is not the full automorphism group of

while the outer automorphism there is an exceptional outer

Symmetric group

49

Conversely, for

has no outer automorphisms, and for

it has no center, so for

it is a

complete group, as discussed in automorphism group, below. For n≥5,

is an almost simple group, as it lies between the simple group

and its group of automorphisms.

Generators and relations The symmetric group on n-letters, Sn, may be described as follows. It has generators:

and relations:

• • • One thinks of

as swapping the i-th and i+1-st position.

Other popular generating sets include the set of transpositions that swap 1 and i for 2 ≤ i ≤ n and any set containing an n-cycle and a 2-cycle.

Subgroup structure A subgroup of a symmetric group is called a permutation group.

Normal subgroups The normal subgroups of the symmetric group are well understood in the finite case. The alternating group of degree n is the only non-identity, proper normal subgroup of the symmetric group of degree n except when n = 1, 2, or 4. In cases n ≤ 2, then the alternating group itself is the identity, but in the case n = 4, there is a second non-identity, proper, normal subgroup, the Klein four group. The normal subgroups of the symmetric groups on infinite sets include both the corresponding "alternating group" on the infinite set, as well as the subgroups indexed by infinite cardinals whose elements fix all but a certain cardinality of elements of the set. For instance, the symmetric group on a countably infinite set has a normal subgroup S consisting of all those permutations which fix all but finitely many elements of the set. The elements of S are each contained in a finite symmetric group, and so are either even or odd. The even elements of S form a characteristic subgroup of S called the alternating group, and are the only other non-identity, proper, normal subgroup of the symmetric group on a countably infinite set. For more details see (Scott 1987, Ch. 11.3) or (Dixon & Mortimer 1996, Ch. 8.1).

Maximal subgroups The maximal subgroups of the finite symmetric groups fall into three classes: the intransitive, the imprimitive, and the primitive. The intransitive maximal subgroups are exactly those of the form Sym(k) × Sym(n−k) for 1 ≤ k < n/2. The imprimitive maximal subgroups are exactly those of the form Sym(k) wr Sym( n/k ) where 2 ≤ k ≤ n/2 is a proper divisor of n and "wr" denotes the wreath product acting imprimitively. The primitive maximal subgroups are more difficult to identify, but with the assistance of the O'Nan–Scott theorem and the classification of finite simple groups, (Liebeck, Praeger & Saxl 1987) gave a fairly satisfactory description of the maximal subgroups of this type according to (Dixon & Mortimer 1996, p. 268).

Symmetric group

50

Sylow subgroups The Sylow subgroups of the symmetric groups are important examples of p-groups. They are more easily described in special cases first: The Sylow p-subgroups of the symmetric group of degree p are just the cyclic subgroups generated by p-cycles. There are (p−1)!/(p−1) = (p−2)! such subgroups simply by counting generators. The normalizer therefore has order p·(p-1) and is known as a Frobenius group Fp(p-1) (especially for p=5), and as the affine general linear group, AGL(1,p). The Sylow p-subgroups of the symmetric group of degree p2 are the wreath product of two cyclic groups of order p. For instance, when p=3, a Sylow 3-subgroup of Sym(9) is generated by a=(1,4,7)(2,5,8)(3,6,9) and the elements x=(1,2,3), y=(4,5,6), z=(7,8,9), and every element of the Sylow 3-subgroup has the form aixjykzl for 0 ≤ i,j,k,l ≤ 2. The Sylow p-subgroups of the symmetric group of degree pn are sometimes denoted Wp(n), and using this notation one has that Wp(n+1) is the wreath product of Wp(n) and Wp(1). In general, the Sylow p-subgroups of the symmetric group of degree n are a direct product of ai copies of Wp(i), where 0 ≤ ai ≤ p−1 and n = a0 + p·a1 + ... + pk·ak. For instance, W2(1) = C2 and W2(2) = D8, the dihedral group of order 8, and so a Sylow 2-subgroup of the symmetric group of degree 7 is generated by { (1,3)(2,4), (1,2), (3,4), (5,6) } and is isomorphic to D8 × C2. These calculations are attributed to (Kaloujnine 1948) and described in more detail in (Rotman 1995, p. 176). Note however that (Kerber 1971, p. 26) attributes the result to an 1844 work of Cauchy, and mentions that it is even covered in textbook form in (Netto 1882, §39–40).

Automorphism group n 1 1

1

1 1

For

,

is a complete group: its center and outer automorphism group are both trivial.

For n = 2, the automorphism group is trivial, but

is not trivial: it is isomorphic to

, which is abelian, and

hence the center is the whole group. For n = 6, it has an outer automorphism of order 2:

, and the automorphism group is a semidirect

product In fact, for any set X of cardinality other than 6, every automorphism of the symmetric group on X is inner, a result first due to (Schreier & Ulam 1937) according to (Dixon & Mortimer 1996, p. 259).

Symmetric group

51

Homology The group homology of

is quite regular and stabilizes: the first homology (concretely, the abelianization) is:

The first homology group is the abelianization, and corresponds to the sign map abelianization for

for

which is the

the symmetric group is trivial. This homology is easily computed as follows:

is generated by involutions (2-cycles, which have order 2), so the only non-trivial maps

are to

and all involutions are conjugate, hence map to the same element in the abelianization (since conjugation is trivial in abelian groups). Thus the only possible maps send an involution to 1 (the trivial map) or to (the sign map). One must also show that the sign map is well-defined, but assuming that, this gives the first homology of The second homology (concretely, the Schur multiplier) is:

This was computed in (Schur 1911), and corresponds to the double cover of the symmetric group, Note that the exceptional low-dimensional homology of the alternating group ( corresponding to non-trivial abelianization, and

due to the exceptional 3-fold cover)

does not change the homology of the symmetric group; the alternating group phenomena do yield symmetric group phenomena – the map extends to and the triple covers of and extend to triple covers of

and

– but these are not homological – the map

does not change the abelianization of

the triple covers do not correspond to homology either. The homology "stabilizes" in the sense of stable homotopy theory: there is an inclusion map fixed k, the induced map on homology

and and for

is an isomorphism for sufficiently high n. This is

analogous to the homology of families Lie groups stabilizing. The homology of the infinite symmetric group is computed in (Nakaoka 1961), with the cohomology algebra forming a Hopf algebra.

Representation theory The representation theory of the symmetric group is a particular case of the representation theory of finite groups, for which a concrete and detailed theory can be obtained. This has a large area of potential applications, from symmetric function theory to problems of quantum mechanics for a number of identical particles. The symmetric group Sn has order n!. Its conjugacy classes are labeled by partitions of n. Therefore according to the representation theory of a finite group, the number of inequivalent irreducible representations, over the complex numbers, is equal to the number of partitions of n. Unlike the general situation for finite groups, there is in fact a natural way to parametrize irreducible representation by the same set that parametrizes conjugacy classes, namely by partitions of n or equivalently Young diagrams of size n. Each such irreducible representation can be realized over the integers (every permutation acting by a matrix with integer coefficients); it can be explicitly constructed by computing the Young symmetrizers acting on a space generated by the Young tableaux of shape given by the Young diagram. Over other fields the situation can become much more complicated. If the field K has characteristic equal to zero or greater than n then by Maschke's theorem the group algebra KSn is semisimple. In these cases the irreducible representations defined over the integers give the complete set of irreducible representations (after reduction modulo the characteristic if necessary).

Symmetric group However, the irreducible representations of the symmetric group are not known in arbitrary characteristic. In this context it is more usual to use the language of modules rather than representations. The representation obtained from an irreducible representation defined over the integers by reducing modulo the characteristic will not in general be irreducible. The modules so constructed are called Specht modules, and every irreducible does arise inside some such module. There are now fewer irreducibles, and although they can be classified they are very poorly understood. For example, even their dimensions are not known in general. The determination of the irreducible modules for the symmetric group over an arbitrary field is widely regarded as one of the most important open problems in representation theory.

See also • • • •

History of group theory Symmetric inverse semigroup Signed symmetric group Generalized symmetric group

References [1] Jacobson (2009), p. 31. [2] Jacobson (2009), p. 32. Theorem 1.1.

• Cameron, Peter J. (1999), Permutation Groups, London Mathematical Society Student Texts, 45, Cambridge University Press, ISBN 978-0-521-65378-7 • Dixon, John D.; Mortimer, Brian (1996), Permutation groups, Graduate Texts in Mathematics, 163, Berlin, New York: Springer-Verlag, MR1409812, ISBN 978-0-387-94599-6 • Jacobson, Nathan (2009), Basic algebra, 1 (2nd ed.), Dover, ISBN 978-0-486-47189-1. • Kaloujnine, Léo (1948), "La structure des p-groupes de Sylow des groupes symétriques finis" (http://www. numdam.org/item?id=ASENS_1948_3_65__239_0), Annales Scientifiques de l'École Normale Supérieure. Troisième Série 65: 239–276, MR0028834, ISSN 0012-9593 • Kerber, Adalbert (1971), Representations of permutation groups. I, Lecture Notes in Mathematics, Vol. 240, 240, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0067943, MR0325752 • Liebeck, M.W.; Praeger, C.E.; Saxl, J. (1988), "On the O'Nan-Scott theorem for finite primitive permutation groups", J. Austral. Math. Soc. 44: 389-396 • Nakaoka, Minoru (March 1961), "Homology of the Infinite Symmetric Group" (http://www.jstor.org/stable/ 1970333), The Annals of Mathematics, 2 (Annals of Mathematics) 73 (2): 229–257, doi:10.2307/1970333 • Netto, E. (1882) (in German), Substitutionentheorie und ihre Anwendungen auf die Algebra., Leipzig. Teubner, JFM 14.0090.01 • Scott, W.R. (1987), Group Theory, New York: Dover Publications, pp. 45–46, ISBN 978-0-486-65377-8 • Schur, Issai (1911), "Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen", Journal für die reine und angewandte Mathematik 139: 155–250 • Schreier, J.; Ulam, Stanislaw (1936), "Über die Automorphismen der Permutationsgruppe der natürlichen Zahlenfolge." (http://matwbn.icm.edu.pl/ksiazki/fm/fm28/fm28128.pdf) (in German), Fundam. Math. 28: 258–260, Zbl: 0016.20301

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Symmetric group

External links • Marcus du Sautoy: Symmetry, reality's riddle (http://www.ted.com/talks/ marcus_du_sautoy_symmetry_reality_s_riddle.html) (video of a talk)

Combinatorial group theory In mathematics, combinatorial group theory is the theory of free groups, and the concept of a presentation of a group, by generators and relations. It is much used in geometric topology, the fundamental group of a simplicial complex having in a natural and geometric way such a presentation. A very closely related topic is geometric group theory, which today largely subsumes combinatorial group theory, using techniques from outside combinatorics besides. It also comprises an number of algorithmically insoluble problems, most notably the word problem for groups; and the classical Burnside problem.

History See (Chandler & Magnus 1982) for a detailed history of combinatorial group theory. A proto-form is found in the 1856 Icosian Calculus of William Rowan Hamilton, where he studied the icosahedral symmetry group via the edge graph of the dodecahedron. The foundations of combinatorial group theory were laid by Walther von Dyck, student of Felix Klein, in the early 1880s, who gave the first systematic study of groups by generators and relations.[1]

References [1] Stillwell, John (2002), Mathematics and its history, Springer, p.  374 (http:/ / books. google. com/ books?id=WNjRrqTm62QC& pg=PA374), ISBN 978-0-38795336-6

• Chandler, B.; Magnus, Wilhelm (December 1, 1982), The History of Combinatorial Group Theory: A Case Study in the History of Ideas, Studies in the History of Mathematics and Physical Sciences (1st ed.), Springer, pp. 234, ISBN 978-0-38790749-9

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Algebraic group

54

Algebraic group In algebraic geometry, an algebraic group (or group variety) is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety. In category theoretic terms, an algebraic group is a group object in the category of algebraic varieties.

Classes Several important classes of groups are algebraic groups, including: • Finite groups • GLnC, the general linear group of invertible matrices over C • Elliptic curves Two important classes of algebraic groups arise, that for the most part are studied separately: abelian varieties (the 'projective' theory) and linear algebraic groups (the 'affine' theory). There are certainly examples that are neither one nor the other — these occur for example in the modern theory of integrals of the second and third kinds such as the Weierstrass zeta function, or the theory of generalized Jacobians. But according to a basic theorem any algebraic group is an extension of an abelian variety by a linear algebraic group. This is a result of Claude Chevalley: if K is a perfect field, and G an algebraic group over K, there exists a unique normal closed subgroup H in G, such that H is a linear group and G/H an abelian variety.[1] According to another basic theorem, any group in the category of affine varieties has a faithful linear representation: we can consider it to be a matrix group over K, defined by polynomials over K and with matrix multiplication as the group operation. For that reason a concept of affine algebraic group is redundant over a field — we may as well use a very concrete definition. Note that this means that algebraic group is narrower than Lie group, when working over the field of real numbers: there are examples such as the universal cover of the 2×2 special linear group that are Lie groups, but have no faithful linear representation. A more obvious difference between the two concepts arises because the identity component of an affine algebraic group G is necessarily of finite index in G. When one wants to work over a base ring R (commutative), there is the group scheme concept: that is, a group object in the category of schemes over R. Affine group scheme is the concept dual to a type of Hopf algebra. There is quite a refined theory of group schemes, that enters for example in the contemporary theory of abelian varieties.

Algebraic subgroup An algebraic subgroup of an algebraic group is a Zariski closed subgroup. Generally these are taken to be connected (or irreducible as a variety) as well. Another way of expressing the condition is as a subgroup which is also a subvariety. This may also be generalized by allowing schemes in place of varieties. The main effect of this in practice, apart from allowing subgroups in which the connected component is of finite index > 1, is to admit non-reduced schemes, in characteristic p.

Coxeter groups There are a number of analogous results between algebraic groups and Coxeter groups – for instance, the number of elements of the symmetric group is , and the number of elements of the general linear group over a finite field is the q-factorial

; thus the symmetric group behaves as though it were a linear group over "the field with one

element". This is formalized by the field with one element, which considers Coxeter groups to be simple algebraic groups over the field with one element.

Algebraic group

Notes [1] Chevalley's result is from 1960 and difficult. Contemporary treatment by Brian Conrad: PDF (http:/ / math. stanford. edu/ ~conrad/ papers/ chev. pdf).

References • Humphreys, James E. (1972), Linear Algebraic Groups, Graduate Texts in Mathematics, 21, Berlin, New York: Springer-Verlag, MR0396773, ISBN 978-0-387-90108-4 • Lang, Serge (1983), Abelian varieties, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90875-5 • Milne, J. S., Algebraic and Arithmetic Groups. (http://www.jmilne.org/math/CourseNotes/AAG.pdf/) • Mumford, David (1970), Abelian varieties, Oxford University Press, ISBN 978-0-19-560528-0, OCLC 138290 • Springer, Tonny A. (1998), Linear algebraic groups, Progress in Mathematics, 9 (2nd ed.), Boston, MA: Birkhäuser Boston, MR1642713, ISBN 978-0-8176-4021-7 • Waterhouse, William C. (1979), Introduction to affine group schemes, Graduate Texts in Mathematics, 66, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90421-4 • Weil, André (1971), Courbes algébriques et variétés abéliennes, Paris: Hermann, OCLC 322901

55

Solvable group

56

Solvable group Concepts in group theory category of groups subgroups, normal subgroups group homomorphisms, kernel, image, quotient direct product, direct sum semidirect product, wreath product Types of groups simple, finite, infinite discrete, continuous multiplicative, additive cyclic, abelian, dihedral nilpotent, solvable list of group theory topics glossary of group theory

In mathematics, more specifically in the field of group theory, a solvable group (or soluble group) is a group that can be constructed from abelian groups using extensions. That is, a solvable group is a group whose derived series terminates in the trivial subgroup. Historically, the word "solvable" arose from Galois theory and the proof of the general unsolvability of quintic equation. Specifically, a polynomial equation is solvable by radicals if and only if the corresponding Galois group is solvable.

Definition A group

is called solvable if it has a subnormal series whose factor groups are all abelian, that is, if there are

subgroups group, for

such that

is normal in

, and

is an abelian

.

Or equivalently, if its derived series, the descending normal series

where every subgroup is the commutator subgroup of the previous one, eventually reaches the trivial subgroup {1} of G. These two definitions are equivalent, since for every group H and every normal subgroup N of H, the quotient H/N is abelian if and only if N includes H(1). The least n such that is called the derived length of the solvable group G. For finite groups, an equivalent definition is that a solvable group is a group with a composition series all of whose factors are cyclic groups of prime order. This is equivalent because a finite abelian group has finite composition length, and every finite simple abelian group is cyclic of prime order. The Jordan–Hölder theorem guarantees that if one composition series has this property, then all composition series will have this property as well. For the Galois group of a polynomial, these cyclic groups correspond to nth roots (radicals) over some field. The equivalence does not necessarily hold for infinite groups: for example, since every nontrivial subgroup of the group Z of integers under addition is isomorphic to Z itself, it has no composition series, but the normal series {0,Z}, with its only factor group isomorphic to Z, proves that it is in fact solvable.

Solvable group In keeping with George Pólya's dictum that "if there's a problem you can't figure out, there's a simpler problem you can figure out", solvable groups are often useful for reducing a conjecture about a complicated group into a conjecture about a series of groups with simple structure: abelian groups (and in the finite case, cyclic groups of prime order).

Examples All abelian groups are solvable - the quotient A/B will always be abelian if A is abelian. But non-abelian groups may or may not be solvable. More generally, all nilpotent groups are solvable. In particular, finite p-groups are solvable, as all finite p-groups are nilpotent. A small example of a solvable, non-nilpotent group is the symmetric group S3. In fact, as the smallest simple non-abelian group is A5, (the alternating group of degree 5) it follows that every group with order less than 60 is solvable. The group S5 is not solvable — it has a composition series {E, A5, S5} (and the Jordan–Hölder theorem states that every other composition series is equivalent to that one), giving factor groups isomorphic to A5 and C2; and A5 is not abelian. Generalizing this argument, coupled with the fact that An is a normal, maximal, non-abelian simple subgroup of Sn for n > 4, we see that Sn is not solvable for n > 4, a key step in the proof that for every n > 4 there are polynomials of degree n which are not solvable by radicals. The celebrated Feit–Thompson theorem states that every finite group of odd order is solvable. In particular this implies that if a finite group is simple, it is either a prime cyclic or of even order. Any finite group whose every p-Sylow subgroups is cyclic is a semidirect product of two cyclic groups, in particular solvable. Such groups are called Z-groups.

Properties Solvability is closed under a number of operations. • If G is solvable, and there is a homomorphism from G onto H, then H is solvable; equivalently (by the first isomorphism theorem), if G is solvable, and N is a normal subgroup of G, then G/N is solvable. • The previous property can be expanded into the following property: G is solvable if and only if both N and G/N are solvable. • If G is solvable, and H is a subgroup of G, then H is solvable. • If G and H are solvable, the direct product G × H is solvable. Solvability is closed under group extension: • If H and G/H are solvable, then so is G; in particular, if N and H are solvable, their semidirect product is also solvable. It is also closed under wreath product: • If G and H are solvable, and X is a G-set, then the wreath product of G and H with respect to X is also solvable. For any positive integer N, the solvable groups of derived length at most N form a subvariety of the variety of groups, as they are closed under the taking of homomorphic images, subalgebras, and (direct) products. The direct product of a sequence of solvable groups with unbounded derived length is not solvable, so the class of all solvable groups is not a variety.

57

Solvable group

Burnside's theorem Burnside's theorem states that if G is a finite group of order

where p and q are prime numbers, and a and b are non-negative integers, then G is solvable.

Related concepts Supersolvable groups As a strengthening of solvability, a group G is called supersolvable (or supersoluble) if it has an invariant normal series whose factors are all cyclic. Since a normal series has finite length by definition, uncountable groups are not supersolvable. In fact, all supersolvable groups are finitely generated, and an abelian group is supersolvable if and only if it is finitely generated. The alternating group A4 is an example of a finite solvable group that is not supersolvable. If we restrict ourselves to finitely generated groups, we can consider the following arrangement of classes of groups: cyclic < abelian < nilpotent < supersolvable < polycyclic < solvable < finitely generated group.

Virtually solvable groups A group G is called virtually solvable if it has a solvable subgroup of finite index. This is similar to virtually abelian. Clearly all solvable groups are virtually solvable, since one can just choose the group itself, which has index 1.

Hypoabelian A solvable group is one whose derived series reaches the trivial subgroup at a finite stage. For an infinite group, the finite derived series may not stabilize, but the transfinite derived series always stabilizes. A group whose transfinite derived series reaches the trivial group is called a hypoabelian group, and every solvable group is a hypoabelian group. The first ordinal α such that G(α) = G(α+1) is called the (transfinite) derived length of the group G, and it has been shown that every ordinal is the derived length of some group (Malcev 1949).

References • Malcev, A. I. (1949), "Generalized nilpotent algebras and their associated groups", Mat. Sbornik N.S. 25 (67): 347–366, MR0032644

External links • Sequence A056866 [1] in the OEIS - orders of non-solvable finite groups.

References [1] http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa056866

58

Solvable subgroup

59

Solvable subgroup Concepts in group theory category of groups subgroups, normal subgroups group homomorphisms, kernel, image, quotient direct product, direct sum semidirect product, wreath product Types of groups simple, finite, infinite discrete, continuous multiplicative, additive cyclic, abelian, dihedral nilpotent, solvable list of group theory topics glossary of group theory

In mathematics, more specifically in the field of group theory, a solvable group (or soluble group) is a group that can be constructed from abelian groups using extensions. That is, a solvable group is a group whose derived series terminates in the trivial subgroup. Historically, the word "solvable" arose from Galois theory and the proof of the general unsolvability of quintic equation. Specifically, a polynomial equation is solvable by radicals if and only if the corresponding Galois group is solvable.

Definition A group

is called solvable if it has a subnormal series whose factor groups are all abelian, that is, if there are

subgroups group, for

such that

is normal in

, and

is an abelian

.

Or equivalently, if its derived series, the descending normal series

where every subgroup is the commutator subgroup of the previous one, eventually reaches the trivial subgroup {1} of G. These two definitions are equivalent, since for every group H and every normal subgroup N of H, the quotient H/N is abelian if and only if N includes H(1). The least n such that is called the derived length of the solvable group G. For finite groups, an equivalent definition is that a solvable group is a group with a composition series all of whose factors are cyclic groups of prime order. This is equivalent because a finite abelian group has finite composition length, and every finite simple abelian group is cyclic of prime order. The Jordan–Hölder theorem guarantees that if one composition series has this property, then all composition series will have this property as well. For the Galois group of a polynomial, these cyclic groups correspond to nth roots (radicals) over some field. The equivalence does not necessarily hold for infinite groups: for example, since every nontrivial subgroup of the group Z of integers under addition is isomorphic to Z itself, it has no composition series, but the normal series {0,Z}, with its only factor group isomorphic to Z, proves that it is in fact solvable.

Solvable subgroup In keeping with George Pólya's dictum that "if there's a problem you can't figure out, there's a simpler problem you can figure out", solvable groups are often useful for reducing a conjecture about a complicated group into a conjecture about a series of groups with simple structure: abelian groups (and in the finite case, cyclic groups of prime order).

Examples All abelian groups are solvable - the quotient A/B will always be abelian if A is abelian. But non-abelian groups may or may not be solvable. More generally, all nilpotent groups are solvable. In particular, finite p-groups are solvable, as all finite p-groups are nilpotent. A small example of a solvable, non-nilpotent group is the symmetric group S3. In fact, as the smallest simple non-abelian group is A5, (the alternating group of degree 5) it follows that every group with order less than 60 is solvable. The group S5 is not solvable — it has a composition series {E, A5, S5} (and the Jordan–Hölder theorem states that every other composition series is equivalent to that one), giving factor groups isomorphic to A5 and C2; and A5 is not abelian. Generalizing this argument, coupled with the fact that An is a normal, maximal, non-abelian simple subgroup of Sn for n > 4, we see that Sn is not solvable for n > 4, a key step in the proof that for every n > 4 there are polynomials of degree n which are not solvable by radicals. The celebrated Feit–Thompson theorem states that every finite group of odd order is solvable. In particular this implies that if a finite group is simple, it is either a prime cyclic or of even order. Any finite group whose every p-Sylow subgroups is cyclic is a semidirect product of two cyclic groups, in particular solvable. Such groups are called Z-groups.

Properties Solvability is closed under a number of operations. • If G is solvable, and there is a homomorphism from G onto H, then H is solvable; equivalently (by the first isomorphism theorem), if G is solvable, and N is a normal subgroup of G, then G/N is solvable. • The previous property can be expanded into the following property: G is solvable if and only if both N and G/N are solvable. • If G is solvable, and H is a subgroup of G, then H is solvable. • If G and H are solvable, the direct product G × H is solvable. Solvability is closed under group extension: • If H and G/H are solvable, then so is G; in particular, if N and H are solvable, their semidirect product is also solvable. It is also closed under wreath product: • If G and H are solvable, and X is a G-set, then the wreath product of G and H with respect to X is also solvable. For any positive integer N, the solvable groups of derived length at most N form a subvariety of the variety of groups, as they are closed under the taking of homomorphic images, subalgebras, and (direct) products. The direct product of a sequence of solvable groups with unbounded derived length is not solvable, so the class of all solvable groups is not a variety.

60

Solvable subgroup

Burnside's theorem Burnside's theorem states that if G is a finite group of order

where p and q are prime numbers, and a and b are non-negative integers, then G is solvable.

Related concepts Supersolvable groups As a strengthening of solvability, a group G is called supersolvable (or supersoluble) if it has an invariant normal series whose factors are all cyclic. Since a normal series has finite length by definition, uncountable groups are not supersolvable. In fact, all supersolvable groups are finitely generated, and an abelian group is supersolvable if and only if it is finitely generated. The alternating group A4 is an example of a finite solvable group that is not supersolvable. If we restrict ourselves to finitely generated groups, we can consider the following arrangement of classes of groups: cyclic < abelian < nilpotent < supersolvable < polycyclic < solvable < finitely generated group.

Virtually solvable groups A group G is called virtually solvable if it has a solvable subgroup of finite index. This is similar to virtually abelian. Clearly all solvable groups are virtually solvable, since one can just choose the group itself, which has index 1.

Hypoabelian A solvable group is one whose derived series reaches the trivial subgroup at a finite stage. For an infinite group, the finite derived series may not stabilize, but the transfinite derived series always stabilizes. A group whose transfinite derived series reaches the trivial group is called a hypoabelian group, and every solvable group is a hypoabelian group. The first ordinal α such that G(α) = G(α+1) is called the (transfinite) derived length of the group G, and it has been shown that every ordinal is the derived length of some group (Malcev 1949).

References • Malcev, A. I. (1949), "Generalized nilpotent algebras and their associated groups", Mat. Sbornik N.S. 25 (67): 347–366, MR0032644

External links • Sequence A056866 [1] in the OEIS - orders of non-solvable finite groups.

61

Tits building

Tits building In mathematics, a building (also Tits building, Bruhat–Tits building, named after François Bruhat and Jacques Tits) is a combinatorial and geometric structure which simultaneously generalizes certain aspects of flag manifolds, finite projective planes, and Riemannian symmetric spaces. Initially introduced by Jacques Tits as a means to understand the structure of exceptional groups of Lie type, the theory has also been used to study the geometry and topology of homogeneous spaces of p-adic Lie groups and their discrete subgroups of symmetries, in the same way that trees have been used to study free groups.

Overview The notion of a building was invented by Jacques Tits as a means of describing simple algebraic groups over an arbitrary field. Tits demonstrated how to every such group G one can associate a simplicial complex Δ = Δ(G) with an action of G, called the spherical building of G. The group G imposes very strong combinatorial regularity conditions on the complexes Δ that can arise in this fashion. By treating these conditions as axioms for a class of simplicial complexes, Tits arrived at his first definition of a building. A part of the data defining a building Δ is a Coxeter group W, which determines a highly symmetrical simplicial complex Σ = Σ(W,S), called the Coxeter complex. A building Δ is glued together from multiple copies of Σ, called its apartments, in a certain regular fashion. When W is a finite Coxeter group, the Coxeter complex is a topological sphere, and the corresponding buildings are said to be of spherical type. When W is an affine Weyl group, the Coxeter complex is a subdivision of the affine plane and one speaks of affine, or Euclidean, buildings. An affine building of type is the same as an infinite tree without terminal vertices. Although the theory of semisimple algebraic groups provided the initial motivation for the notion of a building, not all buildings arise from a group. In particular, projective planes and generalized quadrangles form two classes of graphs studied in incidence geometry which satisfy the axioms of a building, but may not be connected with any group. This phenomenon turns out to be related to the low rank of the corresponding Coxeter system (namely, two). Tits proved a remarkable theorem: all spherical buildings of rank at least three are connected with a group; moreover, if a building of rank at least two is connected with a group then the group is essentially determined by the building. Iwahori–Matsumoto, Borel–Tits and Bruhat–Tits demonstrated that in analogy with Tits' construction of spherical buildings, affine buildings can also be constructed from certain groups, namely, reductive algebraic groups over a local non-Archimedean field. Furthermore, if the split rank of the group is at least three, it is essentially determined by its building. Tits later reworked the foundational aspects of the theory of buildings using the notion of a chamber system, encoding the building solely in terms of adjacency properties of simplices of maximal dimension; this leads to simplifications in both spherical and affine cases. He proved that, in analogy with the spherical case, any building of affine type and rank at least four arises from a group.

Definition An n-dimensional building X is an abstract simplicial complex which is a union of subcomplexes A called apartments such that • every k-simplex of X is contained in an at least three n-simplices if k < n; • any (n – 1 )-simplex in an apartment A lies in exactly two adjacent n-simplices of A and the graph of adjacent n-simplices is connected; • any two simplices in X lie in some common apartment A; • if two simplices both lie in apartments A and A ', then there is a simplicial isomorphism of A onto A ' fixing the vertices of the two simplices.

62

Tits building An n-simplex in A is called a chamber (originally chambre, i.e. room in French). The rank of the building is defined to be n + 1.

Elementary properties Every apartment A in a building is a Coxeter complex. In fact, for every two n-simplices intersecting in an (n – 1)-simplex or panel, there is a unique period two simplicial automorphism of A, called a reflection, carrying one n-simplex onto the other and fixing their common points. These reflections generate a Coxeter group W, called the Weyl group of A, and the simplicial complex A corresponds to the standard geometric realization of W. Standard generators of the Coxeter group are given by the reflections in the walls of a fixed chamber in A. Since the apartment A is determined up to isomorphism by the building, the same is true of any two simplices in X lie in some common apartment A. When W is finite, the building is said to be spherical. When it is an affine Weyl group, the building is said to be affine or euclidean. The chamber system is given by the adjacency graph formed by the chambers; each pair of adjacent chambers can in addition be labelled by one of the standard generators of the Coxeter group (see Tits 1981). Every building has a canonical length metric inherited from the geometric realisation obtained by identifying the vertices with an orthonormal basis of a Hilbert space. For affine buildings, this metric satisfies the CAT(0) comparison inequality of Alexandrov, known in this setting as the Bruhat-Tits non-positive curvature condition for geodesic triangles: the distance from a vertex to the midpoint of the opposite side is no greater than the distance in the corresponding Euclidean triangle with the same side-lengths (see Bruhat & Tits 1972).

Connection with BN pairs If a group G acts simplicially on a building X, transitively on pairs of chambers C and apartments A containing them, then the stabilisers of such a pair define a BN pair or Tits system. In fact the pair of subgroups B = GC and N = GA satisfies the axioms of a BN pair and the Weyl group can identified with N / N B. Conversely the building can be recovered from the BN pair, so that every BN pair canonically defines a building. In fact, using the terminology of BN pairs and calling any conjugate of B a Borel subgroup and any group containing a Borel subgroup a parabolic subgroup, • the vertices of the building X correspond to maximal parabolic subgroups; • k + 1 vertices form a k-simplex whenever the intersection of the corresponding maximal parabolic subgroups is also parabolic; • apartments are conjugates under G of the simplicial subcomplex with vertices given by conjugates under N of maximal parabolics containing B. The same building can often be described by different BN pairs. Moreover not every building comes from a BN pair: this corresponds to the failure of classification results in low rank and dimension (see below).

Spherical and affine buildings for SLn The simplicial structure of the affine and spherical buildings associated to SLn(Qp), as well as their interconnections, are easy to explain directly using only concepts from elementary algebra and geometry (see Garrett 1997). In this case there are three different buildings, two spherical and one affine. Each is a union of apartments, themselves simplicial complexes. For the affine group, an apartment is just the simplicial complex obtained from the standard tessellation of Euclidean space En-1 by equilateral (n-1)-simplices; while for a spherical building it is the finite simplicial complex formed by all (n-1)! simplices with a given common vertex in the analogous tessellation in En-2. Each building is a simplicial complex X which has to satisfy the following axioms:

63

Tits building

64

• X is a union of apartments. • Any two simplices in X are contained in a common apartment. • If a simplex is contained in two apartments, there is a simplicial isomorphism of one onto the other fixing all common points.

Spherical building Let F be a field and let X be the simplicial complex with vertices the non-trivial vector subspaces of V=Fn. Two subspaces U1 and U2 are connected if one of them is a subset of the other. The k-simplices of X are formed by sets of k + 1 mutually connected subspaces. Maximal connectivity is obtained by taking n - 1 subspaces and the corresponding (n-2)-simplex corresponds to a complete flag (0)

U1

···

Un – 1

V

Lower dimensional simplices correspond to partial flags with fewer intermediary subspaces Ui. To define the apartments in X, it is convenient to define a frame in V as a basis (vi) determined up to scalar multiplication of each of its vectors vi; in other words a frame is a set of one-dimensional subspaces Li = F·vi such that any k of them generate a k-dimensional subspace. Now an ordered frame L1, ..., Ln defines a complete flag via Ui = L1

···

Li

Since reorderings of the Li's also give a frame, it is straightforward to see that the subspaces, obtained as sums of the Li's, form a simplicial complex of the type required for an apartment of a spherical building. The axioms for a building can easily be verified using the classical Schreier refinement argument used to prove the uniqueness of the Jordan-Hölder decomposition.

Affine building Let K be a field lying between Q and its p-adic completion Qp with respect to the usual non-Archimedean p-adic norm ||x||p on Q for some prime p. Let R be the subring of K defined by When K = Q, R is the localization of Z at p and, when K = Qp, R = Zp, the p-adic integers, i.e. the closure of Z in Qp. The vertices of the building X are the R-lattices in V = Kn, i.e. R-submodules of the form L = R·v1

···

R·vn

where (vi) is a basis of V over K. Two lattices are said to be equivalent if one is a scalar multiple of the other by an element of the multiplicative group K* of K (in fact only integer powers of p need be used). Two lattice L1 and L2 are said to be adjacent if some lattice equivalent to L2 lies between L1 and its sublattice p·L1: this relation is symmetric. The k-simplices of X are equivalence classes of k + 1 mutually adjacent lattices, The (n - 1)- simplices correspond, after relabelling, to chains p·Ln

L1

L2

···

Ln – 1

Ln

where each successive quotient has order p. Apartments are defined by fixing a basis (vi) of V and taking all lattices with basis (pai vi) where (ai) lies in Zn and is uniquely determined up to addition of the same integer to each entry. By definition each apartment has the required form and their union is the whole of X. The second axiom follows by a variant of the Schreier refinement argument. The last axiom follows by a simple counting argument based on the orders of finite Abelian groups of the form L + pk ·Li / pk ·Li . A standard compactness argument shows that X is in fact independent of the choice of K. In particular taking K = Q, it follows that X is countable. On the other hand taking K = Qp, the definition shows that GLn(Qp) admits a natural simplicial action on the building.

Tits building

65

The building comes equipped with a labelling of its vertices with values in Z / n Z. Indeed, fixing a reference lattice L, the label of M is given by label (M) = logp |M/ pk L| modulo n for k sufficiently large. The vertices of any (n – 1)-simplex in X have distinct labels, running through the whole of Z / n Z. Any simplicial automorphism φ of X defines a permutation π of Z / n Z such that label (φ(M)) = π(label (M)). In particular for g in GLn (Qp), label (g·M) = label (M) + logp || det g ||p modulo n. Thus g preserves labels if g lies in SLn(Qp).

Automorphisms Tits proved that any label-preserving automorphism of the affine building arises from an element of SLn(Qp). Since automorphisms of the building permute the labels, there is a natural homomorphism Aut X

Sn.

The action of GLn(Qp) gives rise to an n-cycle τ. Other automorphisms of the building arise from outer automorphisms of SLn(Qp) associated with automorphisms of the Dynkin diagram. Taking the standard symmetric bilinear form with orthonormal basis vi, the map sending a lattice to its dual lattice gives an automorphism with square the identity, giving the permutation σ that sends each label to its negative modulo n. The image of the above homomorphism is generated by σ and τ and is isomorphic to the dihedral group Dn of order 2n; when n = 3, it gives the whole of S3. If E is a finite Galois extension of Qp and the building is constructed from SLn(E) instead of SLn(Qp), the Galois group Gal (E/Qp) will also act by automorphisms on the building.

Geometric relations Spherical buildings arise in two quite different ways in connection with the affine building X for SLn(Qp): • The link of each vertex L in the affine building corresponds to submodules of L/p·L under the finite field F = R/p·R = Z/(p). This is just the spherical building for SLn(F). • The building X can be compactified by adding the spherical building for SLn(Qp) as boundary "at infinity" (see Garrett 1997 or Brown 1989).

Classification Tits proved that all irreducible spherical buildings (i.e. with finite Weyl group) of rank greater than 2 are associated to simple algebraic or classical groups. A similar result holds for irreducible affine buildings of dimension greater than two (their buildings "at infinity" are spherical of rank greater than two). In lower rank or dimension, there is no such classification. Indeed each incidence structure gives a spherical building of rank 2 (see Pott 1995); and Ballmann and Brin proved that every 2-dimensional simplicial complex in which the links of vertices are isomorphic to the flag complex of a finite projective plane has the structure of a building, not necessarily classical. Many 2-dimensional affine buildings have been constructed using hyperbolic reflection groups or other more exotic constructions connected with orbifolds. Tits also proved that every time a building is described by a BN pair in a group, then in almost all cases the automorphisms of the building correspond to automorphisms of the group (see Tits 1974).

Tits building

Applications The theory of buildings has important applications in several rather disparate fields. Besides the already mentioned connections with the structure of reductive algebraic groups over general and local fields, buildings are used to study their representations. The results of Tits on determination of a group by its building have deep connections with rigidity theorems of George Mostow and Grigory Margulis, and with Margulis arithmeticity. Special types of buildings are studied in discrete mathematics, and the idea of a geometric approach to characterizing simple groups proved very fruitful in the classification of finite simple groups. The theory of buildings of type more general than spherical or affine is still relatively undeveloped, but these generalized buildings have already found applications to construction of Kac-Moody groups in algebra, and to nonpositively curved manifolds and hyperbolic groups in topology and geometric group theory.

See also • • • •

Buekenhout geometry ‎ Coxeter group BN pair Affine Hecke algebra

• • • • • • • •

Bruhat decomposition Generalized polygon Tits geometry Twin building Hyperbolic building Tits simplicity theorem Mostow rigidity Coxeter Complex

References • Ballmann, Werner; Brin, Michael (1995), "Orbihedra of nonpositive curvature" [1], Publications Mathématiques de l'IHÉS 82: 169–209 • Barré, Sylvain (1995), "Polyèdres finis de dimension 2 à courbure ≤ 0 et de rang 2" [2], Ann. Inst. Fourier 45: 1037–1059 • Barré, Sylvain; Pichot, Mikaël (2007), "Sur les immeubles triangulaires et leurs automorphismes" [3], Geom. Dedicata 130: 71–91, doi:10.1007/s10711-007-9206-0 • Bourbaki, Nicolas (1968), Lie Groups and Lie Algebras: Chapters 4-6, Elements of Mathematics, Hermann, ISBN 3-540-42650-7 • Brown, Kenneth S. (1989), Buildings, Springer-Verlag, ISBN 0-387-96876-8 • Bruhat, François; Tits, Jacques (1972), "Groupes réductifs sur un corps local, I. Données radicielles valuées" [4], Publ. Math. IHES 41: 5–251 • Garrett, Paul (1997), Buildings and Classical Groups [5], Chapman & Hall, ISBN 0-412-06331-X • Kantor, William M. (2001), "Tits building" [6], in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104 • Kantor, William M. (1986), "Generalized polygons, SCABs and GABs", in Rosati, L.A., Buildings and the Geometry of Diagrams (CIME Session, Como 1984), Lect. notes in math., 1181, Springer, pp. 79–158, doi:10.1007/BFb0075513 • Pott, Alexander (1995), Finite Geometry and Character Theory, Lect. Notes in Math., 1601, Springer-Verlag, doi:10.1007/BFb0094449, ISBN 354059065X

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Tits building • Ronan, Mark (1995), A construction of buildings with no rank 3 residues of spherical type, Lect. Notes in Math., 1181, Springer-Verlag, pp. 159–190, doi:10.1007/BFb0075518 • Ronan, Mark (1992), "Buildings: main ideas and applications. II. Arithmetic groups, buildings and symmetric spaces", Bull. London Math. Soc. 24 (2): 97–126, doi:10.1112/blms/24.2.97, MR1148671 • Ronan, Mark (1992), "Buildings: main ideas and applications. I. Main ideas.", Bull. London Math. Soc. 24 (1): 1–51, doi:10.1112/blms/24.1.1, MR1139056 • Ronan, Mark (1989), Lectures on buildings, Perspectives in Mathematics 7, Academic Press, ISBN 0-12-594750-X • Tits, Jacques (1974), Buildings of spherical type and finite BN-pairs, Lecture Notes in Mathematics, 386, Springer-Verlag, doi:10.1007/BFb0057391, ISBN 0-387-06757-4 • Tits, Jacques (1981), "A local approach to buildings", The geometric vein: The Coxeter Festschrift, Springer-Verlag, pp. 519–547, ISBN 0387905871 • Tits, Jacques (1986), "Immeubles de type affine", in Rosati, L.A., Buildings and the Geometry of Diagrams (CIME Session, Como 1984), Lect. notes in math., 1181, Springer, pp. 159–190, doi:10.1007/BFb0075514 • Weiss, Richard M. (2003), The structure of spherical buildings, Princeton University Press, ISBN 0-691-11733-0

References [1] [2] [3] [4] [5] [6]

http:/ / www. numdam. org/ item?id=PMIHES_1995__82__169_0 http:/ / www. numdam. org/ numdam-bin/ fitem?id=AIF_1995__45_4_1037_0 http:/ / web. univ-ubs. fr/ lmam/ barre/ henri. pdf http:/ / www. numdam. org/ item?id=PMIHES_1972__41__5_0 http:/ / www. math. umn. edu/ ~garrett/ m/ buildings http:/ / eom. springer. de/ T/ t092900. htm

Finite group In mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of solvable groups and nilpotent groups. A complete determination of the structure of all finite groups is too much to hope for; the number of possible structures soon becomes overwhelming. However, the complete classification of the finite simple groups was achieved, meaning that the "building blocks" from which all finite groups can be built are now known, as each finite group has a composition series. During the second half of the twentieth century, mathematicians such as Chevalley and Steinberg also increased our understanding of finite analogs of classical groups, and other related groups. One such family of groups is the family of general linear groups over finite fields. Finite groups often occur when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving transformations. The theory of Lie groups, which may be viewed as dealing with "continuous symmetry", is strongly influenced by the associated Weyl groups. These are finite groups generated by reflections which act on a finite dimensional Euclidean space. The properties of finite groups can thus play a role in subjects such as theoretical physics and chemistry.

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Finite group

68

Number of groups of a given order Given a positive integer n, it is not at all a routine matter to determine how many isomorphism types of groups of order n there are. Every group of prime order is cyclic, since Lagrange's theorem implies that the cyclic subgroup generated by any of its non-identity elements is the whole group. If n is the square of a prime, then there are exactly two possible isomorphism types of group of order n, both of which are abelian. If n is a higher power of a prime, then results of Graham Higman and Charles Sims give asymptotically correct estimates for the number of isomorphism types of groups of order n, and the number grows very rapidly as the power increases. Depending on the prime factorization of n, some restrictions may be placed on the structure of groups of order n, as a consequence, for example, of results such as the Sylow theorems. For example, every group of order pq is cyclic when q < p are primes with p-1 not divisible by q. For a necessary and sufficient condition, see cyclic number. If n is squarefree, then any group of order n is solvable. A theorem of William Burnside, proved using group characters, states that every group of order n is solvable when n is divisible by fewer than three distinct primes. By the Feit–Thompson theorem, which has a long and complicated proof, every group of order n is solvable when n is odd. For every positive integer n, most groups of order n are solvable. To see this for any particular order is usually not difficult (for example, there is, up to isomorphism, one non-solvable group and 12 solvable groups of order 60) but the proof of this for all orders uses the classification of finite simple groups. For any positive integer n there are at most two simple groups of n, and there are infinitely many positive integers n for which there are two non-isomorphic simple groups of order n.

Table of distinct groups of order n Order n # Groups[1]

Abelian

Non-Abelian

1

1

1

0

2

1

1

0

3

1

1

0

4

2

2

0

5

1

1

0

6

2

1

1

7

1

1

0

8

5

3

2

9

2

2

0

10

2

1

1

11

1

1

0

12

5

2

3

13

1

1

0

14

2

1

1

15

1

1

0

16

14

5

9

17

1

1

0

18

5

2

3

19

1

1

0

Finite group

69 20

5

2

3

21

2

1

1

22

2

1

1

23

1

1

0

24

15

3

12

25

2

2

0

Notes [1] John F. Humphreys, A Course in Group Theory, Oxford University Press, 1996, pp. 238-242.

External references • Number of groups of order n (sequence A000001 (http://en.wikipedia.org/wiki/Oeis:a000001) in OEIS)

p-adic number In mathematics, and chiefly number theory, the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems. The extension is achieved by an alternative interpretation of the concept of absolute value. First described by Kurt Hensel in 1897[1] , the p-adic numbers were motivated primarily by an attempt to bring the ideas and techniques of power series methods into number theory. Their influence now extends far beyond this. For example, the field of p-adic analysis essentially provides an alternative form of calculus. More formally, for a given prime p, the field Qp of p-adic numbers is a completion of the rational numbers. The field Qp is also given a topology derived from a metric, which is itself derived from an alternative valuation on the rational numbers. This metric space is complete in the sense that every Cauchy sequence converges to a point in Qp. This is what allows the development of calculus on Qp, and it is the interaction of this analytic and algebraic structure which gives the p-adic number systems their power and utility. The p in p-adic is a variable and may be replaced with a constant (yielding, for instance, "the 2-adic numbers") or another placeholder variable (for expressions such as "the l-adic numbers").

Introduction This section is an informal introduction to p-adic numbers, using examples from the ring of 10-adic numbers. More formal constructions and properties are given below. In the standard decimal representation, almost all[2] real numbers do not have a terminating decimal representation. For example, 1/3 is represented as a non-terminating decimal as follows

Informally, most people are comfortable with non-terminating decimals because it is clear that a real number can be approximated to any required degree of "closeness" (precision) by a terminating decimal adequately expressed for its intended application. If two decimal expansions differ only after the 10th decimal place they are quite close to one another, and if they differ only after the 20th decimal place they are even closer. 10-adic numbers use a similar non-terminating expansion, but with a different concept of "closeness" (which mathematicians call a metric). Whereas two decimal expansions are close to one another if they differ by a large

''p''-adic number negative power of 10, two 10-adic expansions are close if they differ by a large positive power of 10. Thus 3333 and 4333 are close in the 10-adic metric, and 33333333 and 43333333 are even closer. In the 10-adic metric, the following sequence of numbers gets closer and closer to −1

and taking this sequence to its limit, we can say (informally) that the 10-adic expansion of −1 is

In this notation, 10-adic expansions can be extended indefinitely to the left, in contrast to decimal expansions, which can be extended indefinitely to the right. Note that this is not the only way to write p-adic numbers—for alternatives see the Notation section below. More formally, a 10-adic number can be defined as

where each of the ai is a digit taken from the set {0, 1, …..., 9} and the initial index n may be positive, negative or 0, but must be finite. From this definition, it is clear that positive integers and positive rational numbers with terminating decimal expansions will have terminating 10-adic expansions that are identical to their decimal expansions. Other numbers may have non-terminating 10-adic expansions. It is possible to define addition, subtraction, and multiplication on 10-adic numbers in a consistent way, so that the 10-adic numbers form a commutative ring. We can create 10-adic expansions for negative numbers as follows

and fractions which have non-terminating decimal expansions also have non-terminating 10-adic expansions. For example

Generalizing the last example, we can find a 10-adic expansion for any rational number p⁄q such that q is co-prime to 10; Euler's theorem guarantees that if q is co-prime to 10, then there is an n such that 10n − 1 is a multiple of q. However, 10-adic numbers have one major drawback. It is possible to find pairs of non-zero 10-adic numbers whose product is 0. In other words, the 10-adic numbers are not a domain because they contain zero divisors. This turns out to be because 10 is a composite number. Fortunately, this problem can be avoided by using a prime number p as the base of the number system instead of 10.

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''p''-adic number

p-adic expansions If p is a fixed prime number, then any positive integer can be written in a base p expansion in the form

where the ai are integers in {0, …, p − 1}. For example, the binary expansion of 35 is 1·25 + 0·24 + 0·23 + 0·22 + 1·21 + 1·20, often written in the shorthand notation 1000112. The familiar approach to extending this description to the larger domain of the rationals (and, ultimately, to the reals) is to use sums of the form:

A definite meaning is given to these sums based on Cauchy sequences, using the absolute value as metric. Thus, for example, 1/3 can be expressed in base 5 as the limit of the sequence 0.1313131313...5. In this formulation, the integers are precisely those numbers for which ai = 0 for all i < 0. As an alternative, if we extend the base p expansions by allowing infinite sums of the form

where k is some (not necessarily positive) integer, we obtain the p-adic expansions defining the field Qp of p-adic numbers. Those p-adic numbers for which ai = 0 for all i < 0 are also called the p-adic integers. The p-adic integers form a subring of Qp, denoted Zp. (Not to be confused with the ring of integers modulo p which is also sometimes written Zp. To avoid ambiguity, Z/pZ or Z/(p) are often used to represent the integers modulo p.) Intuitively, as opposed to p-adic expansions which extend to the right as sums of ever smaller, increasingly negative powers of the base p (as is done for the real numbers as described above), these are numbers whose p-adic expansion to the left are allowed to go on forever. For example, the p-adic expansion of 1/3 in base 5 is …1313132, i.e. the limit of the sequence 2, 32, 132, 3132, 13132, 313132, 1313132,… . Multiplying this infinite sum by 3 in base 5 gives …0000001. As there are no negative powers of 5 in this expansion of 1/3 (i.e. no numbers to the right of the decimal point), we see that 1/3 is a p-adic integer in base 5. While it is possible to use this approach to rigorously define p-adic numbers and explore their properties, just as in the case of real numbers other approaches are generally preferred. Hence we want to define a notion of infinite sum which makes these expressions meaningful, and this is most easily accomplished by the introduction of the p-adic metric. Two different but equivalent solutions to this problem are presented in the Constructions section below.

Notation There are several different conventions for writing p-adic expansions. So far this article has used a notation for p-adic expansions in which powers of p increase from right to left. With this right-to-left notation the 3-adic expansion of 1/5, for example, is written as

When performing arithmetic in this notation, digits are carried to the left. It is also possible to write p-adic expansions so that the powers of p increase from left to right, and digits are carried to the right. With this left-to-right notation the 3-adic expansion of 1/5 is

p-adic expansions may be written with other sets of digits instead of {0, 1, …, p − 1}. For example, the 3-adic expansion of 1/5 can be written using balanced ternary digits {1,0,1} as

71

''p''-adic number

72

In fact any set of p integers which are in distinct residue classes modulo p may be used as p-adic digits. In number theory, Teichmüller digits are sometimes used.

Constructions Analytic approach The real numbers can be defined as equivalence classes of Cauchy sequences of rational numbers; this allows us to, for example, write 1 as 1.000… = 0.999… . However, the definition of a Cauchy sequence relies on the metric chosen and, by choosing a different one, numbers other than the real numbers can be constructed. The usual metric which yields the real numbers is called the Euclidean metric. For a given prime p, we define the p-adic absolute value in Q as follows: for any non-zero rational number x, there is a unique integer n allowing us to write x = pn(a/b), where neither of the integers a and b is divisible by p. Unless the numerator or denominator of x in lowest terms contains p as a factor, n will be 0. Now define |x|p = p−n. We also define |0|p = 0. For example with x = 63/550 = 2−1 32 5−2 7 11−1

This definition of |x|p has the effect that high powers of p become "small". By the fundamental theorem of arithmetic, for distinct primes and with for all and , and non-zero integers

and

It now follows that

we can write any non-zero rational number n as follows:

and

for any other prime

It is a theorem of Ostrowski that each absolute value on Q is equivalent either to the Euclidean absolute value, the trivial absolute value, or to one of the p-adic absolute values for some prime p. The p-adic absolute value defines a metric dp on Q by setting The field Qp of p-adic numbers can then be defined as the completion of the metric space (Q,dp); its elements are equivalence classes of Cauchy sequences, where two sequences are called equivalent if their difference converges to zero. In this way, we obtain a complete metric space which is also a field and contains Q. It can be shown that in Qp, every element x may be written in a unique way as

where k is some integer and each ai is in {0, …, p − 1}. This series converges to x with respect to the metric dp. With this absolute value, the field Qp is a local field.

''p''-adic number

73

Algebraic approach In the algebraic approach, we first define the ring of p-adic integers, and then construct the field of fractions of this ring to get the field of p-adic numbers. We start with the inverse limit of the rings Z/pnZ (see modular arithmetic): a p-adic integer is then a sequence (an)n≥1 such that an is in Z/pnZ, and if n < m, an ≡ am (mod pn). Every natural number m defines such a sequence (an) by an = m mod pn and can therefore be regarded as a p-adic integer. For example, in this case 35 as a 2-adic integer would be written as the sequence (1, 3, 3, 3, 3, 35, 35, 35, …). The operators of the ring amount to pointwise addition and multiplication of such sequences. This is well defined because addition and multiplication commute with the mod operator, see modular arithmetic. Moreover, every sequence (an) where the first element is not 0 has an inverse. In that case, for every n, an and p are coprime, and so an and pn are relatively prime. Therefore, each an has an inverse mod pn, and the sequence of these inverses, (bn), is the sought inverse of (an). For example, consider the p-adic integer corresponding to the natural number 7; as a 2-adic number, it would be written (1, 3, 7, 7, 7, 7, 7, ...). This object's inverse would be written as an ever-increasing sequence that begins (1, 3, 7, 7, 23, 55, 55, 183, 439, 439, 1463 ...). Naturally, this 2-adic integer has no corresponding natural number. Every such sequence can alternatively be written as a series of the form we considered above. For instance, in the 3-adics, the sequence (2, 8, 8, 35, 35, ...) can be written as 2 + 2·3 + 0·32 + 1·33 + 0·34 + ... The partial sums of this latter series are the elements of the given sequence. The ring of p-adic integers has no zero divisors, so we can take the field of fractions to get the field Qp of p-adic numbers. Note that in this field of fractions, every non-integer p-adic number can be uniquely written as p−nu with a natural number n and a unit in the p-adic integers u. This means that

Note that

, where

multiplication) of a commutative ring with unit by

is a multiplicative subset (contains the unit and closed under , is an algebraic construction called the ring of fractions of

.

Properties The ring of p-adic integers is the inverse limit of the finite rings Z/pkZ, but is nonetheless uncountable[3] , and has the cardinality of the continuum. Accordingly, the field Qp is uncountable. The endomorphism ring of the Prüfer p-group of rank n, denoted Z(p∞)n, is the ring of n×n matrices over the p-adic integers; this is sometimes referred to as the Tate module. The p-adic numbers contain the rational numbers Q and form a field of characteristic 0. This field cannot be turned into an ordered field. Let the topology τ on Zp be defined by taking as a basis all sets of the form Ua(n) = {n + λ pa for λ in Zp and a in N}. Then Zp is a compactification of Z, under the derived topology (it is not a compactification of Z with its usual topology). The relative topology on Z as a subset of Zp is called the p-adic topology on Z. The topology of the set of p-adic integers is that of a Cantor set; the topology of the set of p-adic numbers is that of a Cantor set minus a point (which would naturally be called infinity)[4] . In particular, the space of p-adic integers is compact while the space of p-adic numbers is not; it is only locally compact. As metric spaces, both the p-adic integers and the p-adic numbers are complete[5] . The real numbers have only a single proper algebraic extension, the complex numbers; in other words, this quadratic extension is already algebraically closed. By contrast, the algebraic closure of the p-adic numbers has infinite degree[6] . Furthermore, Qp has infinitely many inequivalent algebraic extensions. Also contrasting the case of real

''p''-adic number

74

numbers, the algebraic closure of Qp is not (metrically) complete[7] . Its (metric) completion is called Cp. Here an end is reached, as Cp is algebraically closed[8] . The field Cp is isomorphic to the field C of complex numbers, so we may regard Cp as the complex numbers endowed with an exotic metric. It should be noted that the proof of existence of such a field isomorphism relies on the axiom of choice, and does not provide an explicit example of such an isomorphism. The p-adic numbers contain the nth cyclotomic field (n>2) if and only if n divides p − 1[9] . For instance, the nth cyclotomic field is a subfield of Q13 if and only if n = 1, 2, 3, 4, 6, or 12. In particular, there is no multiplicative p-torsion in the p-adic numbers, if p > 2. Also, -1 is the only torsion element in 2-adic numbers. Given a natural number k, the index of the multiplicative group of the k-th powers of the non-zero elements of Qp in the multiplicative group of Qp is finite. The number e, defined as the sum of reciprocals of factorials, is not a member of any p-adic field; but ep is a p-adic number for all p except 2, for which one must take at least the fourth power[10] . (Thus a number with similar properties as e - namely a pth root of ep - is a member of the algebraic closure of the p-adic numbers for all p.) Over the reals, the only functions whose derivative is zero are the constant functions. This is not true over Qp[11] . For instance, the function f: Qp → Qp, f(x) = (1/|x|p)2 for x ≠ 0, f(0) = 0, has zero derivative everywhere but is not even locally constant at 0. Given any elements r∞, r2, r3, r5, r7, ... where rp is in Qp (and Q∞ stands for R), it is possible to find a sequence (xn) in Q such that for all p (including ∞), the limit of xn in Qp is rp. The field Qp is a locally compact Hausdorff space. If

is a finite Galois extension of

, the Galois group

is solvable. Thus, the Galois group

is prosolvable.

Rational arithmetic Hehner and Horspool proposed in 1979 the use of a p-adic representation for rational numbers on computers.[12] The primary advantage of such a representation is that addition, subtraction, and multiplication can be done in a straightforward manner analogous to similar methods for binary integers; and division is even simpler, resembling multiplication. However, it has the disadvantage that representations can be much larger than simply storing the numerator and denominator in binary; for example, if 2n−1 is a Mersenne prime, its reciprocal will require 2n−1 bits to represent.

Generalizations and related concepts The reals and the p-adic numbers are the completions of the rationals; it is also possible to complete other fields, for instance general algebraic number fields, in an analogous way. This will be described now. Suppose D is a Dedekind domain and E is its field of fractions. Pick a non-zero prime ideal P of D. If x is a non-zero element of E, then xD is a fractional ideal and can be uniquely factored as a product of positive and negative powers of non-zero prime ideals of D. We write ordP(x) for the exponent of P in this factorization, and for any choice of number c greater than 1 we can set

Completing with respect to this absolute value |.|P yields a field EP, the proper generalization of the field of p-adic numbers to this setting. The choice of c does not change the completion (different choices yield the same concept of Cauchy sequence, so the same completion). It is convenient, when the residue field D/P is finite, to take for c the size of D/P.

''p''-adic number For example, when E is a number field, Ostrowski's theorem says that every non-trivial non-Archimedean absolute value on E arises as some |.|P. The remaining non-trivial absolute values on E arise from the different embeddings of E into the real or complex numbers. (In fact, the non-Archimedean absolute values can be considered as simply the different embeddings of E into the fields Cp, thus putting the description of all the non-trivial absolute values of a number field on a common footing.) Often, one needs to simultaneously keep track of all the above mentioned completions when E is a number field (or more generally a global field), which are seen as encoding "local" information. This is accomplished by adele rings and idele groups.

Local-global principle Helmut Hasse's local-global principle is said to hold for an equation if it can be solved over the rational numbers if and only if it can be solved over the real numbers and over the p-adic numbers for every prime p.

Notes [1] Hensel, Kurt (1897). "Über eine neue Begründung der Theorie der algebraischen Zahlen" (http:/ / www. digizeitschriften. de/ resolveppn/ GDZPPN00211612X& L=2). Jahresbericht der Deutschen Mathematiker-Vereinigung (http:/ / www. digizeitschriften. de/ resolveppn/ PPN37721857X& L=2) 6 (3): 83–88. . [2] The number of real numbers with terminating decimal representations is countably infinite, while the number of real numbers without such a representation is uncountably infinite. [3] Robert (2000) Section 1.1 [4] Robert (2000) Section 2.3 [5] Gouvêa (2000) Corollary 3.3.8 [6] Gouvêa (2000) Corollary 5.3.10 [7] Gouvêa (2000) Theorem 5.7.4 [8] Gouvêa (2000) Proposition 5.7.8 [9] Gouvêa (2000) Proposition 3.4.2 [10] Robert (2000) Section 4.1 [11] Robert (2000) Section 5.1 [12] Eric C. R. Hehner, R. Nigel Horspool, A new representation of the rational numbers for fast easy arithmetic. SIAM Journal on Computing 8, 124-134. 1979.

References • Gouvêa, Fernando Q. (2000). p-adic Numbers : An Introduction (2nd ed.). Springer. ISBN 3540629114. • Koblitz, Neal (1996). P-adic Numbers, p-adic Analysis, and Zeta-Functions (2nd ed.). Springer. ISBN 0387960171. • Robert, Alain M. (2000). A Course in p-adic Analysis. Springer. ISBN 0387986693. • Bachman, George (1964). Introduction to p-adic Numbers and Valuation Theory. Academic Press. ISBN 0120702681. • Steen, Lynn Arthur (1978). Counterexamples in Topology. Dover. ISBN 048668735X.

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External links • • • •

Weisstein, Eric W., " p-adic Number (http://mathworld.wolfram.com/p-adicNumber.html)" from MathWorld. p-adic integers (http://planetmath.org/?op=getobj&from=objects&id=3118) on PlanetMath p-adic number (http://eom.springer.de/P/p071020.htm) at Springer On-line Encyclopaedia of Mathematics Completion of Algebraic Closure (http://math.stanford.edu/~conrad/248APage/handouts/algclosurecomp. pdf) - on-line lecture notes by Brian Conrad • An Introduction to p-adic Numbers and p-adic Analysis (http://www.maths.gla.ac.uk/~ajb/dvi-ps/ padicnotes.pdf) - on-line lecture notes by Andrew Baker, 2007

Tits alternative In mathematics, the Tits alternative, named for Jacques Tits, is an important theorem about the structure of finitely generated linear groups. It states that every such group is either virtually solvable (i.e. has a solvable subgroup of finite index), or it contains a subgroup isomorphic to the free group on two generators.

Generalization In geometric group theory, a group G is said to satisfy the Tits alternative if for every subgroup H of G either H is virtually solvable or H contains a nonabelian free subgroup (in some versions of the definition this condition is only required to be satisfied for all finitely generated subgroups of G).

References • Tits, J. (1972). "Free subgroups in linear groups". J. Algebra 20: 250–270. doi:10.1016/0021-8693(72)90058-0. • Bestvina, Mladen; Feighn, Mark; Handel, Michael (2000). "The Tits alternative for Out(Fn) I: Dynamics of exponentially-growing automorphisms" [1]. Annals of Mathematics (Annals of Mathematics) 151 (2): 517–623. doi:10.2307/121043.

References [1] http:/ / arxiv. org/ pdf/ math/ 9712217

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Finitely generated group

Finitely generated group In abstract algebra, a generating set of a group is a subset that is not contained in any proper subgroup of the group. Equivalently, a generating set of a group is a subset such that every element of the group can be expressed as the combination (under the group operation) of finitely many elements of the subset and their inverses. More generally, if S is a subset of a group G, then <S>, the subgroup generated by S, is the smallest subgroup of G containing every element of S, meaning the intersection over all subgroups containing the elements of S; equivalently, <S> is the subgroup of all elements of G that can be expressed as the finite product of elements in S and their inverses. If G = <S>, then we say S generates G; and the elements in S are called generators or group generators. If S is the empty set, then <S> is the trivial group {e}, since we consider the empty product to be the identity. When there is only a single element x in S, <S> is usually written as <x>. In this case, <x> is the cyclic subgroup of the powers of x, a cyclic group, and we say this group is generated by x. Equivalent to saying an element x generates a group is saying that <x> equals the entire group G. For finite groups, it is also equivalent to saying that x has order |G|.

Finitely generated group If S is finite, then a group G = <S> is called finitely generated. The structure of finitely generated abelian groups in particular is easily described. Many theorems that are true for finitely generated groups fail for groups in general. It has been proven that if a finite group is generated by a subset S, then each group element may be expressed as a word from the alphabet S of length less than or equal to the order of the group. Every finite group is finitely generated since  = G. The integers under addition are an example of an infinite group which is finitely generated by both <1> and <−1>, but the group of rationals under addition cannot be finitely generated. No uncountable group can be finitely generated. Different subsets of the same group can be generating subsets; for example, if p and q are integers with gcd(p, q) = 1, then <{p, q}> also generates the group of integers under addition (by Bézout's identity). While it is true that every quotient of a finitely generated group is finitely generated (simply take the images of the generators in the quotient), a subgroup of a finitely generated group need not be finitely generated. For example, let G be the free group in two generators, x and y (which is clearly finitely generated, since G = <{x,y}>), and let S be the subset consisting of all elements of G of the form ynxy−n, for n a natural number. Since <S> is clearly isomorphic to the free group in countable generators, it cannot be finitely generated. However, every subgroup of a finitely generated abelian group is in itself finitely generated. Rather more can be said about this though: the class of all finitely generated groups is closed under extensions. To see this, take a generating set for the (finitely generated) normal subgroup and quotient: then the generators for the normal subgroup, together with preimages of the generators for the quotient, generate the group.

77

Finitely generated group

Free group The most general group generated by a set S is the group freely generated by S. Every group generated by S is isomorphic to a factor group of this group, a feature which is utilized in the expression of a group's presentation.

Frattini subgroup An interesting companion topic is that of non-generators. An element x of the group G is a non-generator if every set S containing x that generates G, still generates G when x is removed from S. In the integers with addition, the only non-generator is 0. The set of all non-generators forms a subgroup of G, the Frattini subgroup.

Examples The group of units U(Z9) is the group of all integers relatively prime to 9 under multiplication mod 9 (U9 = {1, 2, 4, 5, 7, 8}). All arithmetic here is done modulo 9. Seven is not a generator of U(Z9), since while 2 is, since:

On the other hand, for n > 2 the symmetric group of degree n is not cyclic, so it is not generated by any one element. However, it is generated by the two permutations (1 2) and (1 2 3 ... n). For example, for S3 we have: e = (1 2)(1 2) (1 2) = (1 2) (1 3) = (1 2)(1 2 3) (2 3) = (1 2 3)(1 2) (1 2 3) = (1 2 3) (1 3 2) = (1 2)(1 2 3)(1 2) Infinite groups can also have finite generating sets. The additive group of integers has 1 as a generating set. The element 2 is not a generating set, as the odd numbers will be missing. The two-element subset {3, 5} is a generating set, since (−5) + 3 + 3 = 1 (in fact, any pair of coprime numbers is, as a consequence of Bézout's identity).

References • Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, 211 (Revised third ed.), New York: Springer-Verlag, MR1878556, ISBN 978-0-387-95385-4

External links • Mathworld: Group generators [1]

References [1] http:/ / mathworld. wolfram. com/ GroupGenerators. html

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Linear group

Linear group In mathematics, a matrix group is a group G consisting of invertible matrices over some field K, usually fixed in advance, with operations of matrix multiplication and inversion. More generally, one can consider n × n matrices over a commutative ring R. (The size of the matrices is restricted to be finite, as any group can be represented as a group of infinite matrices over any field.) A linear group is an abstract group that is isomorphic to a matrix group over a field K, in other words, admitting a faithful, finite-dimensional representation over K. Any finite group is linear, because it can be realized by permutation matrices using Cayley's theorem. Among infinite groups, linear groups form an interesting and tractable class. Examples of groups that are not linear include all "sufficiently large" groups; for example, the infinite symmetric group of permutations of an infinite set.

Basic examples The set MR(n,n) of n × n matrices over a commutative ring R is itself a ring under matrix addition and multiplication. The group of units of MR(n,n) is called the general linear group of n × n matrices over the ring R and is denoted GLn(R) or GL(n,R). All matrix groups are subgroups of some general linear group.

Classical groups Some particularly interesting matrix groups are the so-called classical groups. When the ring of coefficients of the matrix group is the real numbers, these groups are the classical Lie groups. When the underlying ring is a finite field the classical groups are groups of Lie type. These groups play an important role in the classification of finite simple groups.

Finite groups as matrix groups Every finite group is isomorphic to some matrix group. This is similar to Cayley's theorem which states that every finite group is isomorphic to some permutation group. Since the isomorphism property is transitive one need only consider how to form a matrix group from a permutation group. Let G be a permutation group on n points (Ω = {1,2,…,n}) and let {g1,...,gk} be a generating set for G. The general linear group GLn(C) of n×n matrices over the complex numbers acts naturally on the vector space Cn. Let B={b1,…,bn} be the standard basis for Cn. For each gi let Mi in GLn(C) be the matrix which sends each bj to bgi(j). That is, if the permutation gi sends the point j to k then Mi sends the basis vector bj to bk. Let M be the subgroup of GLn(C) generated by {M1,…,Mk}. The action of G on Ω is then precisely the same as the action of M on B. It can be proved that the function taking each gi to Mi extends to an isomorphism and thus every group is isomorphic to a matrix group. Note that the field (C in the above case) is irrelevant since M contains only elements with entries 0 or 1. One can just as easily perform the construction for an arbitrary field since the elements 0 and 1 exist in every field. As an example, let G = S3, the symmetric group on 3 points. Let g1 = (1,2,3) and g2 = (1,2). Then

Notice that M1b1 = b2, M1b2 = b3 and M1b3 = b1. Likewise, M2b1 = b2, M2b2 = b1 and M2b3 = b3.

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Linear group

Representation theory and character theory Linear transformations and matrices are (generally speaking) well-understood objects in mathematics and have been used extensively in the study of groups. In particular representation theory studies homomorphisms from a group into a matrix group and character theory studies homomorphisms from a group into a field given by the trace of a representation.

Examples • See table of Lie groups, list of finite simple groups, and list of simple Lie groups for many examples. • See list of transitive finite linear groups. • In 2000 a longstanding conjecture was resolved when it was shown that the braid groups Bn are linear for all n.[1]

References • Brian C. Hall Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, 1st edition, Springer, 2006. ISBN 0-387-40122-9 • Wulf Rossmann, Lie Groups: An Introduction Through Linear Groups (Oxford Graduate Texts in Mathematics), Oxford University Press ISBN 0-19-859683-9. • La géométrie des groupes classiques, J. Dieudonné. Springer, 1955. ISBN 1-114-75188-X • The classical groups, H. Weyl, ISBN 0-691-05756-7 [1] Stephen J. Bigelow (December 13, 2000), "Braid groups are linear" (http:/ / www. ams. org/ jams/ 2001-14-02/ S0894-0347-00-00361-1/ S0894-0347-00-00361-1. pdf), Journal of the American Mathematical Society 14 (2): 471–486,

External links • Linear groups (http://eom.springer.de/L/l059250.htm), Encyclopaedia of Mathematics

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Finite index

Finite index In mathematics, specifically group theory, the index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" (cosets) of H that fill up G. For example, if H has index 2 in G, then intuitively "half" of the elements of G lie in H. The index of H in G is usually denoted |G : H| or [G : H]. Formally, the index of H in G is defined as the number of cosets of H in G. (The number of left cosets of H in G is always equal to the number of right cosets.) For example, let Z be the group of integers under addition, and let 2Z be the subgroup of Z consisting of the even integers. Then 2Z has two cosets in Z (namely the even integers and the odd integers), so the index of 2Z in Z is two. In general,

for any positive integer n. If N is a normal subgroup of G, then the index of N in G is also equal to the order of the quotient group G / N, since this is defined in terms of a group structure on the set of cosets of N in G. If G is infinite, the index of a subgroup H will in general be a cardinal number. It may however be finite, that is, a positive integer, as the example above shows. If G and H are finite groups, then the index of H in G is equal to the quotient of the orders of the two groups:

This is Lagrange's theorem, and in this case the quotient is necessarily a positive integer.

Properties • If H is a subgroup of G and K is a subgroup of H, then

• If H and K are subgroups of G, then

with equality if HK = G. (If |G : H ∩ K| is finite, then equality holds if and only if HK = G.) • Equivalently, if H and K are subgroups of G, then

with equality if HK = G. (If |H : H ∩ K| is finite, then equality holds if and only if HK = G.) • If G and H are groups and φ: G → H is a homomorphism, then the index of the kernel of φ in G is equal to the order of the image:

• Let G be a group acting on a set X, and let x ∈ X. Then the cardinality of the orbit of x under G is equal to the index of the stabilizer of x:

This is known as the orbit-stabilizer theorem. • As a special case of the orbit-stabilizer theorem, the number of conjugates gxg−1 of an element x ∈ G is equal to the index of the centralizer of x in G. • Similarly, the number of conjugates gHg−1 of a subgroup H in G is equal to the index of the normalizer of H in G. • If H is a subgroup of G, the index of the normal core of H satisfies the following inequality:

where ! denotes the factorial function; this is discussed further below.

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Finite index

82

• As a corollary, if the index of H in G is 2, or for a finite group the lowest prime p that divides the order of G, then H is normal, as the index of its core must also be p, and thus H equals its core, i.e., is normal. • Note that a subgroup of lowest prime index may not exist, such as in any simple group of non-prime order, or more generally any perfect group.

Examples • The alternating group has index 2 in the symmetric group and thus is normal. • The special orthogonal group SO(n) has index 2 in the orthogonal group O(n), and thus is normal. • The free abelian group Z ⊕ Z has three subgroups of index 2, namely . n

n

• More generally, if p is prime then Z has (p  − 1) / (p − 1) subgroups of index p, corresponding to the pn − 1 nontrivial homomorphisms Zn → Z/pZ. • Similarly, the free group Fn has pn − 1 subgroups of index p. • The infinite dihedral group has a cyclic subgroup of index 2, which is necessarily normal.

Infinite index If H has an infinite number of cosets in G, then the index of H in G is said to be infinite. In this case, the index |G : H| is actually a cardinal number. For example, the index of H in G may be countable or uncountable, depending on whether H has a countable number of cosets in G. Note that the index of H is at most the order of G, which is realized for the trivial subgroup, or in fact any subgroup H of infinite cardinality less than that of G.

Finite index An infinite group G may have subgroups H of finite index (for example, the even integers inside the group of integers). Such a subgroup always contains a normal subgroup N (of G), also of finite index. In fact, if H has index n, then the index of N can be taken as some factor of n!. A special case, n = 2, gives the general result that a subgroup of index 2 is a normal subgroup, because the normal group (N above) must have index 2 and therefore be identical to the original subgroup. More generally, a subgroup of index p where p is the smallest prime factor of the order of G (if G is finite) is necessarily normal, as the index of N divides p! and thus must equal p, having no other prime factors. This result is generally proven using group actions; an alternative proof of the result that subgroup of index lowest prime p is normal, and other properties of subgroups of prime index are given in (Lam 2004).

Proof This can be seen more concretely, by considering the permutation action of G on left cosets of H when multiplying them on the right by elements of G (or, equally, multiplying right cosets on the left). This provides a quotient group of G, the image of this permutation representation, which is a subgroup of the symmetric group on n elements. Let us explain this now in more detail. The elements of G that leave all cosets the same form a group. (If Hca ⊂ Hc ∀ c ∈ G and likewise Hcb ⊂ Hc ∀ c ∈ G, then Hcab ⊂ Hc ∀ c ∈ G. If h1ca = h2c for all c ∈ G (with h1, h2 ∈ H) then h2ca−1 = h1c, so Hca−1 ⊂ Hc.) Let us call this group A. Let B be the set of elements of G which perform a given permutation on the cosets of H. Then the cardinality (size) of B is equal to the cardinality of A, and in fact B is a right coset of A. (If cb1 = d and cb2 = hd (a member of the same coset as d), then cb1b2−1 = db2−1 = h−1c ∈ Hc. Since this is the case for any b2 and for any c (with appropriate d), b1b2−1 ∈ A and the size of B is less than or equal to the size of A. Conversely, Hcb1 = Hcab1, and since the left-hand side is in Hd then so is the right-hand side: Hcab1 ⊂ Hcd,

Finite index

83

showing that for any element of A there is a different element of B, and thus the size of A is less than or equal to the size of B.) Since the number of possible permutations of cosets is finite, namely n! (assuming H is of finite index n), then there can only be a finite number of sets like B. If G is infinite, then all such sets are therefore infinite. The set of these sets forms a group isomorphic to a subset of the group of permutations, so the number of these sets must divide n!. Finally, if for some c ∈ G and a ∈ A we have ca = xc, then for any d ∈ G dca = hdc for some h ∈ H, but also dca = dxc, so hd = dx. Since this is true for any d, x must be a member of A, so ca = xc implies that A is a normal subgroup.

Examples The above considerations are true for finite groups as well. For instance, the group O of chiral octahedral symmetry has 24 elements. It has a dihedral D4 subgroup (in fact it has three such) of order 8, and thus of index 3 in O, which we shall call H. This dihedral group has a 4-member D2 subgroup, which we may call A. Multiplying on the right any element of a right coset of H by an element of A gives a member of the same coset of H (Hca = Hc). A is normal in O. There are six cosets of A, corresponding to the six elements of the symmetric group S3. All elements from any particular coset of A perform the same permutation of the cosets of H. On the other hand, the group Th of pyritohedral symmetry also has 24 members and a subgroup of index 3 (this time it is a D2h prismatic symmetry group, see point groups in three dimensions), but in this case the whole subgroup is a normal subgroup. All members of a particular coset carry out the same permutation of these cosets, but in this case they represent only the 3-element alternating group in the 6-member S3 symmetric group.

Normal subgroups of prime power index Normal subgroups of prime power index are kernels of surjective maps to p-groups and have interesting structure, as described at Focal subgroup theorem: Subgroups and elaborated at focal subgroup theorem. There are three important normal subgroups of prime power index, each being the smallest normal subgroup in a certain class: • Ep(G) is the intersection of all index p normal subgroups; G/Ep(G) is an elementary abelian group, and is the largest elementary abelian p-group onto which G surjects. • Ap(G) is the intersection of all normal subgroups K such that G/K is an abelian p-group (i.e., K is an index normal subgroup that contains the derived group

): G/Ap(G) is the largest abelian p-group (not

necessarily elementary) onto which G surjects. • Op(G) is the intersection of all normal subgroups K of G such that G/K is a (possibly non-abelian) p-group (i.e., K is an index normal subgroup): G/Op(G) is the largest p-group (not necessarily abelian) onto which G surjects. Op(G) is also known as the p-residual subgroup. As these are weaker conditions on the groups K, one obtains the containments These groups have important connections to the Sylow subgroups and the transfer homomorphism, as discussed there.

Finite index

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Geometric structure An elementary observation is that one cannot have exactly 2 subgroups of index 2, as their symmetric difference yields a third. This is a simple corollary of the above discussion (namely the projectivization of the vector space structure of the elementary abelian group ), and further, G does not act on this geometry, nor does it reflect any of the non-abelian structure (in both cases because the quotient is abelian). However, it is an elementary result, which can be seen concretely as follows: the set of normal subgroups of a given index p form a projective space, namely the projective space In detail, the space of homomorphisms from G to the (cyclic) group of order p, over the finite field

is a vector space

A non-trivial such map has as kernel a normal subgroup of index p, and

multiplying the map by an element of

(a non-zero number mod p) does not change the kernel; thus one

obtains a map from Conversely, a normal subgroup of index p determines a non-trivial map to

to normal index p subgroups. up to a choice of "which coset maps

to which shows that this map is a bijection. As a consequence, the number of normal subgroups of index p is some k;

for

corresponds to no normal subgroups of index p. Further, given two distinct normal subgroups of

index p, one obtains a projective line consisting of such subgroups. For the symmetric difference of two distinct index 2 subgroups (which are necessarily normal) gives the third point on the projective line containing these subgroups, and a group must contain

index 2

subgroups – it cannot contain exactly 2 or 4 index 2 subgroups, for instance.

References • Lam, T. Y. (March 2004), "On Subgroups of Prime Index" (http://www.jstor.org/stable/4145135), The American Mathematical Monthly 111 (3): 256–258, alternative download (http://math.berkeley.edu/~lam/ html/index-p.ps)

External links • Normality of subgroups of prime index (http://planetmath.org/encyclopedia/ NormalityOfSubgroupsOfPrimeIndex.html) at PlanetMath. • " Subgroup of least prime index is normal (http://groupprops.subwiki.org/wiki/ Subgroup_of_least_prime_index_is_normal)" at Groupprops, The Group Properties Wiki (http://groupprops. subwiki.org/wiki/Main_Page)

Free subgroup

Free subgroup In mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many elements of S and their inverses (disregarding trivial variations such as st−1 = su−1ut−1). Apart from the existence of inverses no other relation exists between the generators of a free group. A related but different notion is a free abelian group.

History Free groups first arose in the study of hyperbolic geometry, as examples of Fuchsian groups (discrete groups acting by isometries on the hyperbolic plane). In an 1882 paper, Walther von Dyck pointed out The Cayley graph for the free group on two that these groups have the simplest possible presentations.[1] The generators. Each vertex represents an element of the free group, and each edge represents algebraic study of free groups was initiated by Jakob Nielsen in 1924, multiplication by a or b. who gave them their name and established many of their basic properties.[2] [3] [4] Max Dehn realized the connection with topology, and obtained the first proof of the full Nielsen-Schreier Theorem.[5] Otto Schreier published an algebraic proof of this result in 1927,[6] and Kurt Reidemeister included a comprehensive treatment of free groups in his 1932 book on combinatorial topology.[7] Later on in the 1930s, Wilhelm Magnus discovered the connection between the lower central series of free groups and free Lie algebras.

Examples The group (Z,+) of integers is free; we can take S = {1}. A free group on a two-element set S occurs in the proof of the Banach–Tarski paradox and is described there. On the other hand, any nontrivial finite group cannot be free, since the elements of a free generating set of a free group have infinite order. In algebraic topology, the fundamental group of a bouquet of k circles (a set of k loops having only one point in common) is the free group on a set of k elements.

Construction The free group FS with free generating set S can be constructed as follows. S is a set of symbols and we suppose for every s in S there is a corresponding "inverse" symbol, s−1, in a set S−1. Let T = S ∪ S−1, and define a word in S to be any written product of elements of T. That is, a word in S is an element of the monoid generated by T. The empty word is the word with no symbols at all. For example, if S = {a, b, c}, then T = {a, a−1, b, b−1, c, c−1}, and

is a word in S. If an element of S lies immediately next to its inverse, the word may be simplified by omitting the s, s−1 pair:

A word that cannot be simplified further is called reduced. The free group FS is defined to be the group of all reduced words in S. The group operation in FS is concatenation of words (followed by reduction if necessary). The identity is the empty word. A word is called cyclically reduced, if its first and last letter are not inverse to each other. Every word is conjugate to a cyclically reduced word, and the cyclically reduced conjugates of a cyclically

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Free subgroup reduced word are all cyclic permutations. For instance b−1abcb is not cyclically reduced, but is conjugate to abc, which is cyclically reduced. The only cyclically reduced conjugates of abc are abc, bca, and cab.

Universal property The free group FS is the universal group generated by the set S. This can be formalized by the following universal property: given any function ƒ from S to a group G, there exists a unique homomorphism φ: FS → G making the following diagram commute:

That is, homomorphisms FS → G are in one-to-one correspondence with functions S → G. For a non-free group, the presence of relations would restrict the possible images of the generators under a homomorphism. To see how this relates to the constructive definition, think of the mapping from S to FS as sending each symbol to a word consisting of that symbol. To construct φ for given ƒ, first note that φ sends the empty word to identity of G and it has to agree with ƒ on the elements of S. For the remaining words (consisting of more than one symbol) φ can be uniquely extended since it is a homomorphism, i.e., φ(ab) = φ(a) φ(b). The above property characterizes free groups up to isomorphism, and is sometimes used as an alternative definition. It is known as the universal property of free groups, and the generating set S is called a basis for FS. The basis for a free group is not uniquely determined. Being characterized by a universal property is the standard feature of free objects in universal algebra. In the language of category theory, the construction of the free group (similar to most constructions of free objects) is a functor from the category of sets to the category of groups. This functor is left adjoint to the forgetful functor from groups to sets.

Facts and theorems Some properties of free groups follow readily from the definition: 1. Any group G is the homomorphic image of some free group F(S). Let S be a set of generators of G. The natural map f: F(S) → G is an epimorphism, which proves the claim. Equivalently, G is isomorphic to a quotient group of some free group F(S). The kernel of f is a set of relations in the presentation of G. If S can be chosen to be finite here, then G is called finitely generated. 2. If S has more than one element, then F(S) is not abelian, and in fact the center of F(S) is trivial (that is, consists only of the identity element). 3. Two free groups F(S) and F(T) are isomorphic if and only if S and T have the same cardinality. This cardinality is called the rank of the free group F. Thus for every cardinal number k, there is, up to isomorphism, exactly one free group of rank k. 4. A free group of finite rank n > 1 has an exponential growth rate of order 2n − 1. A few other related results are: 1. The Nielsen–Schreier theorem: Any subgroup of a free group is free. 2. A free group of rank k clearly has subgroups of every rank less than k. Less obviously, a free group of rank greater than 1 has subgroups of all countable ranks.

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Free subgroup 3. The commutator subgroup of a free group of rank k > 1 has infinite rank; for example for F(a,b), it is freely generated by the commutators [am, bn] for non-zero m and n. 4. The free group in two elements is SQ universal; the above follows as any SQ universal group has subgroups of all countable ranks. 5. Any group that acts on a tree, freely and preserving the orientation, is a free group of countable rank (given by 1 plus the Euler characteristic of the quotient graph). 6. The Cayley graph of a free group of finite rank, with respect to a free generating set, is a tree on which the group acts freely, preserving the orientation. 7. The groupoid approach to these results, given in the work by P.J. Higgins below, is kind of extracted from an approach using covering spaces. It allows more powerful results, for example on Grushko's theorem, and a normal form for the fundamental groupoid of a graph of groups. In this approach there is considerable use of free groupoids on a directed graph. 8. Grushko's theorem has the consequence that if a subset B of a free group F on n elements generates F and has n elements, then B generates F freely.

Free abelian group The free abelian group on a set S is defined via its universal property in the analogous way, with obvious modifications: Consider a pair (F, φ), where F is an abelian group and φ: S → F is a function. F is said to be the free abelian group on S with respect to φ if for any abelian group G and any function ψ: S → G, there exists a unique homomorphism f: F → G such that f(φ(s)) = ψ(s), for all s in S. The free abelian group on S can be explicitly identified as the free group F(S) modulo the subgroup generated by its commutators, [F(S), F(S)], i.e. its abelianisation. In other words, the free abelian group on S is the set of words that are distinguished only up to the order of letters. The rank of a free group can therefore also be defined as the rank of its abelianisation as a free abelian group.

Tarski's problems Around 1945, Alfred Tarski asked whether the free groups on two or more generators have the same first order theory, and whether this theory is decidable. Sela (2006) answered the first question by showing that any two nonabelian free groups have the same first order theory, and Kharlampovich & Myasnikov (2006) answered both questions, showing that this theory is decidable. A similar unsolved (in 2008) question in free probability theory asks whether the von Neumann group algebras of any two non-abelian finitely generated free groups are isomorphic.

Notes [1] von Dyck, Walther (1882). "Gruppentheoretische Studien" (http:/ / www. springerlink. com/ content/ t8lx644qm87p3731). Mathematische Annalen 20 (1): 1–44. doi:10.1007/BF01443322. . [2] Nielsen, Jakob (1917). "Die Isomorphismen der allgemeinen unendlichen Gruppe mit zwei Erzeugenden" (http:/ / www. springerlink. com/ content/ xp12702q30q40381). Mathematische Annalen 78 (1): 385–397. doi:10.1007/BF01457113. MR1511907, JFM 46.0175.01. . [3] Nielsen, Jakob (1921). "On calculation with noncommutative factors and its application to group theory. (Translated from Danish)". The Mathematical Scientist 6 (1981) (2): 73–85. [4] Nielsen, Jakob (1924). "Die Isomorphismengruppe der freien Gruppen" (http:/ / www. springerlink. com/ content/ l898u32j37u10671). Mathematische Annalen 91 (3): 169–209. doi:10.1007/BF01556078. . [5] See Magnus, Wilhelm; Moufang, Ruth (1954). "Max Dehn zum Gedächtnis" (http:/ / www. springerlink. com/ content/ l657774u3w864mp3). Mathematische Annalen 127 (1): 215–227. doi:10.1007/BF01361121. .. [6] Schreier, Otto (1928). "Die Untergruppen der freien Gruppen". Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 5: 161–183. doi:10.1007/BF02952517. [7] Reidemeister, Kurt (1972 (1932 original)). Einführung in die kombinatorische Topologie. Darmstadt: Wissenschaftliche Buchgesellschaft.

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References • Kharlampovich, Olga; Myasnikov, Alexei (2006). "Elementary theory of free non-abelian groups". J. Algebra 302 (2): 451–552. doi:10.1016/j.jalgebra.2006.03.033. MR2293770 • W. Magnus, A. Karrass and D. Solitar, "Combinatorial Group Theory", Dover (1976). • P.J. Higgins, 1971, "Categories and Groupoids", van Nostrand, {New York}. Reprints in Theory and Applications of Categories, 7 (2005) pp 1–195. • Sela, Z. (2006). "Diophantine geometry over groups. VI. The elementary theory of a free group.". Geom. Funct. Anal. 16 (3): 707–730. MR2238945 • J.-P. Serre, Trees, Springer (2003) (English translation of "arbres, amalgames, SL2", 3rd edition, astérisque 46 (1983)) • P.J. Higgins, "The fundamental groupoid of a graph of groups", J. London Math. Soc. (2) {13}, (1976) 145–149. • Aluffi, Paolo (2009). Algebra: Chapter 0 (http://books.google.com/books?id=deWkZWYbyHQC&pg=PA70). AMS Bookstore. p. 70. ISBN 978-0-821-84781-7. • Grillet, Pierre (2007). Abstract algebra (http://books.google.com/books?id=LJtyhu8-xYwC&pg=PA27). Springer. p. 27. ISBN 978-0-387-71567-4.

Tits group The Tits group 2F4(2)′ is a finite simple group of order 17971200 = 211 · 33 · 52 · 13 found by Jacques Tits (1964). The Ree groups 2F4(22n+1) were constructed by Ree (1961), who showed that they are simple if n≥1. The first member of this series 2F4(2) is not simple. It was studied by Jacques Tits (1964) who showed that its derived subgroup 2F4(2)′ of index 2 was a new simple group. The group 2F4(2) is a group of Lie type and has a BN pair, but the Tits group does not, so is strictly speaking not a group of Lie type, though it is usually classed with the groups of Lie type in lists of simple groups as it is so close to one.

Properties The Schur multiplier of the Tits group is trivial and its outer automorphism group has order 2, with the full automorphism group being the group 2F4(2). The group 2F4(2) occurs as a maximal subgroup of the Rudvalis group, as the point stabilizer of the rank 3 permuation action on 4060 = 1+1755+2304 points. Wilson (1984) and Tchakerian (1986) independently found the 8 classes of maximal subgroup of the Tits group. The Tits group is one of the simple N-groups, and was overlooked in John Thompson's first announcement of the classification of simple N-groups, as it had not been discovered at the time. It is also one of the thin finite groups. The Tits group was characterized in various ways by Parrott (1972, 1973) and Stroth (1980).

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Tits group

89

Presentation The Tits group can be defined in terms of generators and relations by

where [a,b] is the commutator. (a,bbabababababbababababa).

It

has

an

outer

automorphism

obtained

by

sending

(a,b)

to

References • Parrott, David (1972), "A characterization of the Tits' simple group" [1], Canadian Journal of Mathematics 24: 672–685, MR0325757, ISSN 0008-414X • Parrott, David (1973), "A characterization of the Ree groups 2F4(q)", Journal of Algebra 27: 341–357, doi:10.1016/0021-8693(73)90109-9, MR0347965, ISSN 0021-8693 • Ree, Rimhak (1961), "A family of simple groups associated with the simple Lie algebra of type (F4)" [2], Bulletin of the American Mathematical Society 67: 115–116, doi:10.1090/S0002-9904-1961-10527-2, MR0125155, ISSN 0002-9904 • Stroth, Gernot (1980), "A general characterization of the Tits simple group" [3], Journal of Algebra 64 (1): 140–147, doi:10.1016/0021-8693(80)90138-6, MR575787, ISSN 0021-8693 • Tchakerian, Kerope B. (1986), "The maximal subgroups of the Tits simple group", Pliska Studia Mathematica Bulgarica 8: 85–93, MR866648, ISSN 0204-9805 • Tits, Jacques (1964), "Algebraic and abstract simple groups" [4], Annals of Mathematics. Second Series 80: 313–329, MR0164968, ISSN 0003-486X • Wilson, Robert A. (1984), "The geometry and maximal subgroups of the simple groups of A. Rudvalis and J. Tits" [5], Proceedings of the London Mathematical Society. Third Series 48 (3): 533–563, doi:10.1112/plms/s3-48.3.533, MR735227, ISSN 0024-6115

External links • ATLAS of Group Representations — The Tits Group [6]

References [1] [2] [3] [4] [5] [6]

http:/ / books. google. com/ books?id=TY5tZCQcK1IC& pg=PA672 http:/ / www. ams. org/ journals/ bull/ 1961-67-01/ S0002-9904-1961-10527-2/ home. html http:/ / dx. doi. org/ 10. 1016/ 0021-8693(80)90138-6 http:/ / www. jstor. org/ stable/ 1970394 http:/ / dx. doi. org/ 10. 1112/ plms/ s3-48. 3. 533 http:/ / brauer. maths. qmul. ac. uk/ Atlas/ v3/ exc/ TF42/

Tits–Koecher construction

Tits–Koecher construction In algebra, the Kantor–Koecher–Tits construction is a method of constructing a Lie algebra from a Jordan algebra, introduced by Jacques Tits (1962), Kantor (1964), and Koecher (1967). If J is a Jordan algebra, the Kantor–Koecher–Tits construction puts a Lie algebra structure on J + J + J + Inner(J), the sum of 3 copies of J and the Lie algebra of inner derivations of J. When applies to a 27-dimensional exceptional Jordan algebra it gives a Lie algebra of type E7 of dimension 133. The Kantor–Koecher–Tits construction was used by Kac (1977) to classify the finite dimensional simple Jordan superalgebras.

References • Jacobson, Nathan (1968), Structure and representations of Jordan algebras, American Mathematical Society Colloquium Publications, Vol. XXXIX, Providence, R.I.: American Mathematical Society, MR0251099 • Kac, Victor G (1977), "Classification of simple Z-graded Lie superalgebras and simple Jordan superalgebras", Communications in Algebra 5 (13): 1375–1400, doi:10.1080/00927877708822224, MR0498755, ISSN 0092-7872 • Kantor, I. L. (1964), "Classification of irreducible transitive differential groups", Doklady Akademii Nauk SSSR 158: 1271–1274, MR0175941, ISSN 0002-3264 • Koecher, Max (1967), "Imbedding of Jordan algebras into Lie algebras. I" [1], American Journal of Mathematics 89: 787–816, MR0214631, ISSN 0002-9327 • Tits, Jacques (1962), "Une classe d'algèbres de Lie en relation avec les algèbres de Jordan", Nederl. Akad. Wetensch. Proc. Ser. A 65 = Indagationes Mathematicae 24: 530–535, MR0146231

References [1] http:/ / www. jstor. org/ stable/ 2373242

90

Primitive group

91

Primitive group In mathematics, a permutation group G acting on a set X is called primitive if G acts transitively on X and G preserves no nontrivial partition of X. In the other case, G is imprimitive. An imprimitive permutation group is an example of an induced representation; examples include coset representations G/H in cases where H is not a maximal subgroup. When H is maximal, the coset representation is primitive. If the set X is finite, its cardinality is called the "degree" of G. The numbers of primitive groups of small degree were stated by Robert Carmichael in 1937: Degree

2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

Number 1 2 2 5 4 7 7 11 9

8

6

9

4

6

22 10 4

8

4

Note the large number of primitive groups of degree 16. As Carmichael notes, all of these groups, except for the symmetric and alternating group, are subgroups of the affine group on the 4-dimensional space over the 2-element finite field. The number of primitive permutation groups of degree n, for n = 0, 1, … , is recorded as sequence A000019 [1] in the On-Line Encyclopedia of Integer Sequences. While primitive permutation groups are transitive by definition, not all transitive permutation groups are primitive.

Examples • Consider the symmetric group

acting on the set

and the permutation

. The group generated by

is primitive.

• Now consider the symmetric group

acting on the set

and the permutation

. The group generated by

is not primitive, since the partition

is preserved under

, i.e.

and

where

and

.

See also • Block (permutation group theory)

References • Roney-Dougal, Colva M. The primitive permutation groups of degree less than 2500, Journal of Algebra 292 (2005), no. 1, 154–183. • The GAP [2] Data Library "Primitive Permutation Groups" [3]. • Carmichael, Robert D., Introduction to the Theory of Groups of Finite Order. Ginn, Boston, 1937. Reprinted by Dover Publications, New York, 1956. • Rowland, Todd; Primitive Group Action. MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. [4]

Primitive group

References [1] [2] [3] [4]

http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa000019 http:/ / www. gap-system. org http:/ / www. gap-system. org/ Datalib/ prim. html http:/ / mathworld. wolfram. com/ PrimitiveGroupAction. html

Geometric group theory Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act (that is, when the groups in question are realized as geometric symmetries or continuous transformations of some spaces). Another important idea in geometric group theory is to consider finitely generated groups themselves as geometric objects. This is usually done by studying the Cayley graphs of groups, which, in addition to the graph structure, are endowed with the structure of a metric space, given by the so-called word metric. Geometric group theory, as a distinct area, is relatively new, and has become a clearly identifiable branch of mathematics in late 1980s and early 1990s. Geometric group theory closely interacts with low-dimensional topology, hyperbolic geometry, algebraic topology, computational group theory and geometric analysis. There are also substantial connections with complexity theory, mathematical logic, the study of Lie Groups and their discrete subgroups, dynamical systems, probability theory, K-theory, and other areas of mathematics. In the introduction to his book Topics in Geometric Group Theory, Pierre de la Harpe wrote: "One of my personal beliefs is that fascination with symmetries and groups is one way of coping with frustrations of life's limitations: we like to recognize symmetries which allow us to recognize more than what we can see. In this sense the study of geometric group theory is a part of culture, and reminds me of several things that Georges de Rham practices on many occasions, such as teaching mathematics, reciting Mallarmé, or greeting a friend" (page 3 in [1] ).

Historical background Geometric group theory grew out of combinatorial group theory that largely studied properties of discrete groups via analyzing group presentations, that describe groups as quotients of free groups; this field was first systematically studied by Walther von Dyck, student of Felix Klein, in the early 1880s,[2] while an early form is found in the 1856 Icosian Calculus of William Rowan Hamilton, where he studied the icosahedral symmetry group via the edge graph of the dodecahedron. Currently combinatorial group theory as an area is largely subsumed by geometric group theory. Moreover, the term "geometric group theory" came to often include studying discrete groups using probabilistic, measure-theoretic, arithmetic, analytic and other approaches that lie outside of the traditional combinatorial group theory arsenal. In the first half of the 20th century, pioneering work of Dehn, Nielsen, Reidemeister and Schreier, Whitehead, van Kampen, amongst others, introduced some topological and geometric ideas into the study of discrete groups.[3] Other precursors of geometric group theory include small cancellation theory and Bass–Serre theory. Small cancellation theory was introduced by Martin Grindlinger in 1960s[4] [5] and further developed by Roger Lyndon and Paul Schupp.[6] It studies van Kampen diagrams, corresponding to finite group presentations, via combinatorial curvature conditions and derives algebraic and algorithmic properties of groups from such analysis. Bass–Serre theory, introduced in the 1977 book of Serre,[7] derives structural algebraic information about groups by studying group actions on simplicial trees. External precursors of geometric group theory include the study of lattices in Lie Groups, especially Mostow rigidity theorem, the study of Kleinian groups, and the progress achieved in low-dimensional topology and hyperbolic geometry in 1970s and early 1980s, spurred, in particular, by Thurston's Geometrization

92

Geometric group theory program. The emergence of geometric group theory as a distinct area of mathematics is usually traced to late 1980s and early 1990s. It was spurred by the 1987 monograph of Gromov "Hyperbolic groups"[8] that introduced the notion of a hyperbolic group (also known as word-hyperbolic or Gromov-hyperbolic or negatively curved group), which captures the idea of a finitely generated group having large-scale negative curvature, and by his subsequent monograph Asymptotic Invariants of Inifinite Groups,[9] that outlined Gromov's program of understanding discrete groups up to quasi-isometry. The work of Gromov had a transformative effect on the study of discrete groups[10] [11] [12] and the phrase "geometric group theory" started appearing soon afterwards. (see, e.g.,[13] ).

Notable themes and developments in geometric group theory Notable themes and developments in geometric group theory in 1990s and 2000s include: • Gromov's program to study quasi-isometric properties of groups. A particularly influential broad theme in the area is Gromov's program[14] of classifying finitely generated groups according to their large scale geometry. Formally, this means classifying finitely generated groups with their word metric up to quasi-isometry. This program involves: 1. The study of properties that are invariant under quasi-isometry. Examples of such properties of finitely generated groups include: the growth rate of a finitely generated group; the isoperimetric function or Dehn function of a finitely presented group; the number of ends of a group; hyperbolicity of a group; the homeomorphism type of the boundary of a hyperbolic group;[15] asymptotic cones of finitely generated groups (see, e.g.,[16] [17] ); amenability of a finitely generated group; being virtually abelian (that is, having an abelian subgroup of finite index); being virtually nilpotent; being virtually free; being finitely presentable; being a finitely presentable group with solvable Word Problem; and others. 2. Theorems which use quasi-isometry invariants to prove algebraic results about groups, for example: Gromov's polynomial growth theorem; Stallings' ends theorem; Mostow rigidity theorem. 3. Quasi-isometric rigidity theorems, in which one classifies algebraically all groups that are quasi-isometric to some given group or metric space. This direction was initiated by the work of Schwartz on quasi-isometric rigidity of rank-one lattices[18] and the work of Farb and Mosher on quasi-isometric rigidity of Baumslag-Solitar groups.[19] • The theory of word-hyperbolic and relatively hyperbolic groups. A particularly important development here is the work of Sela in 1990s resulting in the solution of the isomorphism problem for word-hyperbolic groups.[20] The notion of a relatively hyperbolic groups was originally introduced by Gromov in 1987[8] and refined by Farb[21] and Bowditch,[22] in the 1990s. The study of relatively hyperbolic groups gained prominence in 2000s. • Interactions with mathematical logic and the study of first-order theory of free groups. Particularly important progress occurred on the famous Tarski conjectures, due to the work of Sela[23] as well as of Kharlampovich and Myasnikov.[24] The study of limit groups and introduction of the language and machinery of non-commutative algebraic geometry gained prominence. • Interactions with computer science, complexity theory and the theory of formal languages. This theme is exemplified by the development of the theory of automatic groups,[25] a notion that imposes certain geometric and language theoretic conditions on the multiplication operation in a finitely generate group. • The study of isoperimetric inequalities, Dehn functions and their generalizations for finitely presented group. This includes, in particular, the work of Birget, Ol'shanskii, Rips and Sapir[26] [27] essentially characterizing the possible Dehn functions of finitely presented groups, as well as results providing explicit constructions of groups with fractional Dehn functions.[28] • Development of the theory of JSJ-decompositions for finitely generated and finitely presented groups.[29] [30] [31] [32] [33]

93

Geometric group theory • Connections with geometric analysis, the study of

• • • • •

94 -algebras associated with discrete groups and of the theory

of free probability. This theme is represented, in particular, by considerable progress on the Novikov conjecture and the Baum-Connes conjecture and the development and study of related group-theoretic notions such as topological amenability, asymptotic dimension, uniform embeddability into Hilbert spaces, rapid decay property, and so on (see, for example,[34] [35] [36] ). Interactions with the theory of quasiconformal analysis on metric spaces, particularly in relation to Cannon's Conjecture about characterization of hyperbolic groups with boundary homeomorphic to the 2-sphere.[37] [38] [39] Interactions with topological dynamics in the contexts of studying actions of discrete groups on various compact spaces and group compactifications, particularly convergence group methods[40] [41] Development of the theory of group actions on -trees (particularly the Rips machine), and its applications.[42] The study of group actions on CAT(0) spaces and CAT(0) cubical complexes,[43] motivated by ideas from Alexandrov geometry. Interactions with low-dimensional topology and hyperbolic geometry, particularly the study of 3-manifold groups (see, e.g.,[44] ), mapping class groups of surfaces, braid groups and Kleinian groups.

• Introduction of probabilistic methods to study algebraic properties of "random" group theoretic objects (groups, group elements, subgroups, etc.). A particularly important development here is the work of Gromov who used probabilistic methods to prove[45] the existence of a finitely generated group that is not uniformly embeddable into a Hilbert space. Other notable developments include introduction and study of the notion of generic-case complexity[46] for group-theoretic and other mathematical algorithms and algebraic rigidity results for generic groups.[47] • The study of automata groups and iterated monodromy groups as groups of automorphisms of infinite rooted trees. In particular, Grigorchuk's groups of intermediate growth, and their generalizations, appear in this context.[48] [49] • The study of measure-theoretic properties of group actions on measure spaces, particularly introduction and development of the notions of measure equivalence and orbit equivalence, as well as measure-theoretic generalizations of Mostow rigidity.[50] [51] • The study of unitary representations of discrete groups and Kazhdan's property (T)[52] • The study of Out(Fn) (the outer automorphism group of a free group of rank n) and of individual automorphisms of free groups. Introduction and the study of Culler-Vogtmann's outer space[53] and of the theory of train tracks[54] for free group automorphisms played a particularly prominent role here. • Development of Bass–Serre theory, particularly various accessibility results[55] [56] [57] and the theory of tree lattices.[58] Generalizations of Bass–Serre theory such as the theory of complexes of groups.[59] • The study of random walks on groups and related boundary theory, particularly the notion of Poisson boundary (see, e.g.,[60] ). The study of amenability and of groups whose amenability status is still unknown. • Interactions with finite group theory, particularly progress in the study of subgroup growth.[61] • Studying subgroups and lattices in linear groups, such as

, and of other Lie Groups, via geometric

methods (e.g. buildings), algebro-geometric tools (e.g. algebraic groups and representation varieties), analytic methods (e.g. unitary representations on Hilbert spaces) and arithmetic methods. • Group cohomology, using algebraic and topological methods, particularly involving interaction with algebraic topology and the use of morse-theoretic ideas in the combinatorial context; large-scale, or coarse (e.g. see [62] ) homological and cohomological methods. • Progress on traditional combinatorial group theory topics, such as the Burnside problem,[63] [64] the study of Coxeter groups and Artin groups, and so on (the methods used to study these questions currently are often geometric and topological).

Geometric group theory

Examples The following examples are often studied in geometric group theory: • • • • • • • • • • • • • • •

Amenable groups Free Burnside groups The infinite cyclic group Z Free groups Free products Outer automorphism groups Out(Fn) (via Outer space) Hyperbolic groups Mapping class groups (automorphisms of surfaces) Symmetric groups Braid groups Coxeter groups General Artin groups Thompson's group F CAT(0) groups Arithmetic groups

• • • • • •

Automatic groups Kleinian groups, and other lattices acting on symmetric spaces. Wallpaper groups Baumslag-Solitar groups Fundamental groups of graphs of groups Grigorchuk group

References [1] P. de la Harpe, Topics in geometric group theory. (http:/ / books. google. com/ books?id=60fTzwfqeQIC& pg=PP1& dq=de+ la+ Harpe,+ Topics+ in+ geometric+ group+ theory) Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 2000. ISBN 0-226-31719-6; 0-226-31721-8. [2] Stillwell, John (2002), Mathematics and its history, Springer, p.  374 (http:/ / books. google. com/ books?id=WNjRrqTm62QC& pg=PA374), ISBN 978-0-38795336-6 [3] Bruce Chandler and Wilhelm Magnus. The history of combinatorial group theory. A case study in the history of ideas. Studies in the History of Mathematics and Physical Sciences, vo. 9. Springer-Verlag, New York, 1982. [4] M. Greendlinger, Dehn's algorithm for the word problem. (http:/ / www3. interscience. wiley. com/ journal/ 113397463/ abstract?CRETRY=1& SRETRY=0) Communications in Pure and Applied Mathematics, vol. 13 (1960), pp. 67-83. [5] M. Greendlinger, An analogue of a theorem of Magnus. Archiv der Mathematik, vol. 12 (1961), pp. 94-96. [6] R. Lyndon and P. Schupp, Combinatorial Group Theory (http:/ / books. google. com/ books?id=aiPVBygHi_oC& printsec=frontcover& dq=lyndon+ and+ schupp), Springer-Verlag, Berlin, 1977. Reprinted in the "Classics in mathematics" series, 2000. [7] J.-P. Serre, Trees. Translated from the 1977 French original by John Stillwell. Springer-Verlag, Berlin-New York, 1980. ISBN 3-540-10103-9. [8] M. Gromov, Hyperbolic Groups, in "Essays in Group Theory" (G. M. Gersten, ed.), MSRI Publ. 8, 1987, pp. 75-263. [9] M. Gromov, "Asymptotic invariants of infinite groups", in "Geometric Group Theory", Vol. 2 (Sussex, 1991), London Mathematical Society Lecture Note Series, 182, Cambridge University Press, Cambridge, 1993, pp. 1-295. [10] I. Kapovich and N. Benakli. Boundaries of hyperbolic groups. Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001), pp. 39-93, Contemp. Math., 296, Amer. Math. Soc., Providence, RI, 2002. From the Introduction:" In the last fifteen years geometric group theory has enjoyed fast growth and rapidly increasing influence. Much of this progress has been spurred by remarkable work of M. L. Gromov [in Essays in group theory, 75--263, Springer, New York, 1987; in Geometric group theory, Vol. 2 (Sussex, 1991), 1--295, Cambridge Univ. Press, Cambridge, 1993], who has advanced the theory of word-hyperbolic groups (also referred to as Gromov-hyperbolic or negatively curved groups)." [11] B. H. Bowditch, Hyperbolic 3-manifolds and the geometry of the curve complex. European Congress of Mathematics, pp. 103-115, Eur. Math. Soc., Zürich, 2005. From the Introduction:" Much of this can be viewed in the context of geometric group theory. This subject has seen very rapid growth over the last twenty years or so, though of course, its antecedents can be traced back much earlier. [...] The work of Gromov

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Geometric group theory has been a major driving force in this. Particularly relevant here is his seminal paper on hyperbolic groups [Gr]." [12] G. Elek. The mathematics of Misha Gromov. Acta Mathematica Hungarica, vol. 113 (2006), no. 3, pp. 171-185. From p. 181: "Gromov's pioneering work on the geometry of discrete metric spaces and his quasi-isometry program became the locomotive of geometric group theory from the early eighties." [13] Geometric group theory. Vol. 1. Proceedings of the symposium held at Sussex University, Sussex, July 1991. Edited by Graham A. Niblo and Martin A. Roller. London Mathematical Society Lecture Note Series, 181. Cambridge University Press, Cambridge, 1993. ISBN 0-521-43529-3. [14] M. Gromov, Asymptotic invariants of infinite groups, in "Geometric Group Theory", Vol. 2 (Sussex, 1991), London Mathematical Society Lecture Note Series, 182, Cambridge University Press, Cambridge, 1993, pp. 1-295. [15] I. Kapovich and N. Benakli. Boundaries of hyperbolic groups. Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001), pp. 39-93, Contemp. Math., 296, Amer. Math. Soc., Providence, RI, 2002. [16] T. R. Riley, Higher connectedness of asymptotic cones. (http:/ / www. sciencedirect. com/ science?_ob=ArticleURL& _udi=B6V1J-48173YV-2& _user=10& _rdoc=1& _fmt=& _orig=search& _sort=d& view=c& _acct=C000050221& _version=1& _urlVersion=0& _userid=10& md5=836106f8cf958990dfd27ab111c1286a) Topology, vol. 42 (2003), no. 6, pp. 1289-1352. [17] L. Kramer, S. Shelah, K. Tent and S. Thomas. Asymptotic cones of finitely presented groups. (http:/ / www. sciencedirect. com/ science?_ob=ArticleURL& _udi=B6W9F-4CSG3HS-1& _user=10& _rdoc=1& _fmt=& _orig=search& _sort=d& view=c& _acct=C000050221& _version=1& _urlVersion=0& _userid=10& md5=6ba86760e3a9331e0b330a291a0cf444) Advances in Mathematics, vol. 193 (2005), no. 1, pp. 142-173. [18] R. E. Richard. The quasi-isometry classification of rank one lattices. Institut des Hautes Études Scientifiques. Publications Mathématiques. No. 82 (1995), pp. 133-168. [19] B. Farb and L. Mosher. A rigidity theorem for the solvable Baumslag-Solitar groups. With an appendix by Daryl Cooper. Inventiones Mathematicae, vol. 131 (1998), no. 2, pp. 419-451. [20] Z. Sela, The isomorphism problem for hyperbolic groups. I. (http:/ / www. jstor. org/ pss/ 2118520) Annals of Mathematics (2), vol. 141 (1995), no. 2, pp. 217-283. [21] B. Farb. Relatively hyperbolic groups. Geometric and Functional Analysis, vol. 8 (1998), no. 5, pp. 810-840. [22] B. H. Bowditch. Treelike structures arising from continua and convergence groups. Memoirs American Mathematical Society vol. 139 (1999), no. 662. [23] Z.Sela, Diophantine geometry over groups and the elementary theory of free and hyperbolic groups. Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), pp. 87-92, Higher Ed. Press, Beijing, 2002. [24] O. Kharlampovich and A. Myasnikov, Tarski's problem about the elementary theory of free groups has a positive solution. Electronic Research Announcements of the American Mathematical Society, vol. 4 (1998), pp. 101-108. [25] D. B. A. Epstein, J. W. Cannon, D. Holt, S. Levy, M. Paterson, W. Thurston. Word processing in groups. Jones and Bartlett Publishers, Boston, MA, 1992. [26] M. Sapir, J.-C. Birget, E. Rips, Isoperimetric and isodiametric functions of groups. Annals of Mathematics (2), vol 156 (2002), no. 2, pp. 345-466. [27] J.-C. Birget, A. Yu. Ol'shanskii, E. Rips, M. Sapir, Isoperimetric functions of groups and computational complexity of the word problem. Annals of Mathematics (2), vol 156 (2002), no. 2, pp. 467-518. [28] M. R. Bridson, Fractional isoperimetric inequalities and subgroup distortion. Journal of the American Mathematical Society, vol. 12 (1999), no. 4, pp. 1103-1118. [29] E. Rips and Z. Sela, Cyclic splittings of finitely presented groups and the canonical JSJ decomposition. Annals of Mathematics (2), vol. 146 (1997), no. 1, pp. 53-109. [30] M. J. Dunwoody and M. E. Sageev. JSJ-splittings for finitely presented groups over slender groups. Inventiones Mathematicae, vol. 135 (1999), no. 1, pp. 25-44. [31] P. Scott and G. A. Swarup. Regular neighbourhoods and canonical decompositions for groups. Electronic Research Announcements of the American Mathematical Society, vol. 8 (2002), pp. 20-28. [32] B. H. Bowditch. Cut points and canonical splittings of hyperbolic groups. Acta Mathematica, vol. 180 (1998), no. 2, pp. 145-186. [33] K. Fujiwara and P. Papasoglu, JSJ-decompositions of finitely presented groups and complexes of groups. Geometric and Functional Analysis, vol. 16 (2006), no. 1, pp. 70-125. [34] G. Yu. The Novikov conjecture for groups with finite asymptotic dimension. Annals of Mathematics (2), vol. 147 (1998), no. 2, pp. 325-355. [35] G. Yu. The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space. Inventiones Mathematicae, vol 139 (2000), no. 1, pp. 201--240. [36] I. Mineyev and G. Yu. The Baum-Connes conjecture for hyperbolic groups. Inventiones Mathematicae, vol. 149 (2002), no. 1, pp. 97-122. [37] M. Bonk and B. Kleiner. Conformal dimension and Gromov hyperbolic groups with 2-sphere boundary. Geometry and Topology, vol. 9 (2005), pp. 219-246. [38] M. Bourdon and H. Pajot. Quasi-conformal geometry and hyperbolic geometry. Rigidity in dynamics and geometry (Cambridge, 2000), pp. 1-17, Springer, Berlin, 2002. [39] M. Bonk, Quasiconformal geometry of fractals. International Congress of Mathematicians. Vol. II, pp. 1349-1373, Eur. Math. Soc., Zürich, 2006.

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Geometric group theory [40] P. Tukia. Generalizations of Fuchsian and Kleinian groups. First European Congress of Mathematics, Vol. II (Paris, 1992), pp. 447-461, Progr. Math., 120, Birkhäuser, Basel, 1994. [41] A. Yaman. A topological charactesization of relatively hyperbolic groups. Journal für die Reine und Angewandte Mathematik, vol. 566 (2004), pp. 41-89. [42] M. Bestvina and M. Feighn. Stable actions of groups on real trees. Inventiones Mathematicae, vol. 121 (1995), no. 2, pp. 287-321. [43] M. R. Bridson and A. Haefliger, Metric spaces of non-positive curvature. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319. Springer-Verlag, Berlin, 1999. [44] M. Kapovich, Hyperbolic manifolds and discrete groups. Progress in Mathematics, 183. Birkhäuser Boston, Inc., Boston, MA, 2001. [45] M. Gromov. Random walk in random groups. Geometric and Functional Analysis, vol. 13 (2003), no. 1, pp. 73-146. [46] I. Kapovich, A. Miasnikov, P. Schupp and V. Shpilrain, Generic-case complexity, decision problems in group theory, and random walks. Journal of Algebra, vol. 264 (2003), no. 2, pp. 665-694. [47] I. Kapovich, P. Schupp, V. Shpilrain, Generic properties of Whitehead's algorithm and isomorphism rigidity of random one-relator groups. Pacific Journal of Mathematics, vol. 223 (2006), no. 1, pp. 113-140. [48] L. Bartholdi, R. I. Grigorchuk and Z. Sunik. Branch groups. Handbook of algebra, Vol. 3, pp. 989-1112, North-Holland, Amsterdam, 2003. [49] V. Nekrashevych. Self-similar groups. Mathematical Surveys and Monographs, 117. American Mathematical Society, Providence, RI, 2005. ISBN 0-8218-3831-8. [50] A. Furman, Gromov's measure equivalence and rigidity of higher rank lattices. Annals of Mathematics (2), vol. 150 (1999), no. 3, pp. 1059-1081. [51] N. Monod, Y. Shalom, Orbit equivalence rigidity and bounded cohomology. Annals of Mathematics (2), vol. 164 (2006), no. 3, pp. 825-878. [52] Y. Shalom. The algebraization of Kazhdan's property (T). International Congress of Mathematicians. Vol. II, pp. 1283-1310, Eur. Math. Soc., Zürich, 2006. [53] M Culler and K. Vogtmann. Moduli of graphs and automorphisms of free groups. Inventiones Mathematicae, vol. 84 (1986), no. 1, pp. 91-119. [54] M. Bestvina and M. Handel, Train tracks and automorphisms of free groups. Annals of Mathematics (2), vol. 135 (1992), no. 1, pp. 1-51. [55] M. J. Dunwoody. The accessibility of finitely presented groups. Inventiones Mathematicae, vol. 81 (1985), no. 3, pp. 449-457. [56] M. Bestvina and M. Feighn. Bounding the complexity of simplicial group actions on trees. Inventiones Mathematicae, vol. 103 (1991), no 3, pp. 449-469 (1991). [57] Z. Sela, Acylindrical accessibility for groups. Inventiones Mathematicae, vol. 129 (1997), no. 3, pp. 527-565. [58] H. Bass and A. Lubotzky. Tree lattices. With appendices by Bass, L. Carbone, Lubotzky, G. Rosenberg and J. Tits. Progress in Mathematics, 176. Birkhäuser Boston, Inc., Boston, MA, 2001. ISBN 0-8176-4120-3. [59] M. R. Bridson and A. Haefliger, Metric spaces of non-positive curvature. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319. Springer-Verlag, Berlin, 1999. ISBN 3-540-64324-9. [60] V. A. Kaimanovich, The Poisson formula for groups with hyperbolic properties. Annals of Mathematics (2), vol. 152 (2000), no. 3, pp. 659-692. [61] A. Lubotzky and D. Segal. Subgroup growth. Progress in Mathematics, 212. Birkhäuser Verlag, Basel, 2003. ISBN 3-7643-6989-2. [62] M. Bestvina, M. Kapovich and B. Kleiner. Van Kampen's embedding obstruction for discrete groups. Inventiones Mathematicae, vol. 150 (2002), no. 2, pp. 219-235. [63] S. V. Ivanov. The free Burnside groups of sufficiently large exponents. International Journal of Algebra and Computation, vol. 4 (1994), no. 1-2. [64] I. G. Lysënok. Infinite Burnside groups of even period. (Russian) Izvestial Rossiyskoi Akademii Nauk Seriya Matematicheskaya, vol. 60 (1996), no. 3, pp. 3-224; translation in Izvestiya. Mathematics vol. 60 (1996), no. 3, pp. 453-654.

Books and monographs on or closely related to geometric group theory • B. H. Bowditch. A course on geometric group theory. MSJ Memoirs, 16. Mathematical Society of Japan, Tokyo, 2006. ISBN 4-931469-35-3 • M. R. Bridson and A. Haefliger, Metric spaces of non-positive curvature. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319. Springer-Verlag, Berlin, 1999. ISBN 3-540-64324-9 • P. de la Harpe, Topics in geometric group theory. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 2000. ISBN 0-226-31719-6 • D. B. A. Epstein, J. W. Cannon, D. Holt, S. Levy, M. Paterson, W. Thurston. Word processing in groups. Jones and Bartlett Publishers, Boston, MA, 1992. ISBN 0-86720-244-0 • M. Gromov, Hyperbolic Groups, in "Essays in Group Theory" (G. M. Gersten, ed.), MSRI Publ. 8, 1987, pp. 75–263. ISBN 0-387-96618-8

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• M. Gromov, Asymptotic invariants of infinite groups, in "Geometric Group Theory", Vol. 2 (Sussex, 1991), London Mathematical Society Lecture Note Series, 182, Cambridge University Press, Cambridge, 1993, pp. 1–295 • M. Kapovich, Hyperbolic manifolds and discrete groups. Progress in Mathematics, 183. Birkhäuser Boston, Inc., Boston, MA, 2001 • R. Lyndon and P. Schupp, Combinatorial Group Theory, Springer-Verlag, Berlin, 1977. Reprinted in the "Classics in mathematics" series, 2000. ISBN 3-540-41158-5 • A. Yu. Ol'shanskii, Geometry of defining relations in groups. Translated from the 1989 Russian original by Yu. A. Bakhturin. Mathematics and its Applications (Soviet Series), 70. Kluwer Academic Publishers Group, Dordrecht, 1991 • J. Roe, Lectures on coarse geometry. University Lecture Series, 31. American Mathematical Society, Providence, RI, 2003. ISBN 0-8218-3332-4

External links • John McCammond's Geometric Group Theory Page (http://www.math.ucsb.edu/~mccammon/ geogrouptheory/) • What is Geometric Group Theory? By Daniel Wise (http://www.math.mcgill.ca/wise/ggt/cayley.html) • Open Problems in combinatorial and geometric group theory (http://zebra.sci.ccny.cuny.edu/web/nygtc/ problems/) • Geometric group theory Theme on arxiv.org (http://xstructure.inr.ac.ru/x-bin/theme3.py?level=1& index1=-98867)

Hyperbolic group In group theory, a hyperbolic group, also known as a word hyperbolic group, Gromov hyperbolic group, negatively curved group is a finitely generated group equipped with a word metric satisfying certain properties characteristic of hyperbolic geometry. The notion of a hyperbolic group was introduced and developed by Mikhail Gromov in the early 1980s. He noticed that many results of Max Dehn concerning the fundamental group of a hyperbolic Riemann surface do not rely either on it having dimension two or even on being a manifold and hold in much more general context. In a very influential paper from 1987, Gromov proposed a wide-ranging research program. Ideas and foundational material in the theory of hyperbolic groups also stem from the work of George Mostow, William Thurston, James W. Cannon, Eliyahu Rips, and many others.

Examples of hyperbolic groups • • • •

Finite groups. Virtually cyclic groups. Finitely generated free groups, and more generally, groups that act on a locally finite tree with finite stabilizers. Most surface groups are hyperbolic, namely, the fundamental groups of surfaces with negative Euler characteristic. For example, the fundamental group of the sphere with two handles (the surface of genus two) is a hyperbolic group.

• Most triangle groups

are hyperbolic, namely, those for which 1/l + 1/m + 1/n < 1, such as the (2,3,7)

triangle group. • The fundamental groups of compact Riemannian manifolds with strictly negative sectional curvature. • Groups that act cocompactly and properly discontinuously on a proper CAT(k) space with k < 0. This class of groups includes all the preceding ones as special cases. It also leads to many examples of hyperbolic groups not

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related to trees or manifolds.

Examples of non-hyperbolic groups • The free rank 2 abelian group Z2 is not hyperbolic. • More generally, any group which contains Z2 as a subgroup is not hyperbolic.[1] In particular, lattices in higher rank semisimple Lie groups and the fundamental groups π1(S3−K) of nontrivial knot complements fall into this category and therefore are not hyperbolic. • Baumslag–Solitar groups B(m,n) and any group that contains a subgroup isomorphic to some B(m,n) fail to be hyperbolic (since B(1,1) = Z2, this generalizes the previous example). • A non-uniform lattice in rank 1 semisimple Lie groups is hyperbolic if and only if the associated symmetric space is the hyperbolic plane.

Definitions Hyperbolic groups can be defined in several different ways. All definitions use the Cayley graph of the group and involve a choice of a positive constant and first define a -hyperbolic group. A group is called hyperbolic if it is -hyperbolic for some

. When translating between different definitions of hyperbolicity, the particular value of

may change, but the resulting notions of a hyperbolic group turn out to be equivalent. Let G be a finitely generated group, and T be its Cayley graph with respect to some finite set S of generators. By identifying each edge isometrically with the unit interval in R, the Cayley graph becomes a metric space. The group G acts on T by isometries and this action is simply transitive on the vertices. A path in T of minimal length that connects points x and y is called a geodesic segment and is denoted [x,y]. A geodesic triangle in T consists of three points x, y, z, its vertices, and three geodesic segments [x,y], [y,z], [z,x], its sides. The first approach to hyperbolicity is based on the slim triangles condition and is generally credited to Rips. Let be fixed. A geodesic triangle is -slim if each side is contained in a -neighborhood of the other two sides: The Cayley graph T is

-hyperbolic if all geodesic triangles are

-slim, and in this case G is a

-hyperbolic

group. Although a different choice of a finite generating set will lead to a different Cayley graph and hence to a different condition for G to be -hyperbolic, it is known that the notion of hyperbolicity, for some value of is actually independent of the generating set. In the language of metric geometry, it is invariant under quasi-isometries. Therefore, the property of being a hyperbolic group depends only on the group itself.

Remark By imposing the slim triangles condition on geodesic metric spaces in general, one arrives at the more general notion of -hyperbolic space. Hyperbolic groups can be characterized as groups G which admit an isometric properly discontinuous action on a proper geodesic Δ-hyperbolic space X such that the factor-space X/G has finite diameter.

Homological characterization In 2002, I. Mineyev showed that hyperbolic groups are exactly those finitely generated groups for which the comparison map between the bounded cohomology and ordinary cohomology is surjective in all degrees, or equivalently, in degree 2.

Hyperbolic group

Properties Hyperbolic groups have a soluble word problem. They are biautomatic and automatic.[2] : indeed, they are strongly geodesically automatic, that is, there is an automatic structure on the group, where the language accepted by the word acceptor is the set of all geodesic words. In a 2010 paper[3] , it was shown that hyperbolic groups have a decidable marked isomorphism problem. It is notable that this means that the isomorphism problem, orbit problems (in particular the conjugacy problem) and Whitehead's problem are all decidable.

Generalizations An important generalization of hyperbolic groups in geometric group theory is the notion of a relatively hyperbolic group. Motivating examples for this generalization are given by the fundamental groups of non-compact hyperbolic manifolds of finite volume, in particular, the fundamental groups of hyperbolic knots, which are not hyperbolic in the sense of Gromov. A group G is relatively hyperbolic with respect to a subgroup H if, after contracting the Cayley graph of G along H-cosets, the resulting graph equipped with the usual graph metric is a δ-hyperbolic space and, moreover, it satisfies an additional technical condition which implies that quasi-geodesics with common endpoints travel through approximately the same collection of cosets and enter and exit these cosets in approximately the same place.

Notes [1] Ghys and de la Harpe, Ch. 8, Th. 37; Bridson and Haefliger, Chapter 3.Γ, Corollary 3.10. [2] Charney, Ruth (1992), "Artin groups of finite type are biautomatic", Mathematische Annalen 292: 671–683, doi:10.1007/BF01444642 [3] Dahmani, F.; Guirardel, V. - On the Isomorphism Problem in all Hyperbolic Groups, arXiV: 1002.2590 (http:/ / arxiv. org/ abs/ 1002. 2590)

References • Mikhail Gromov, Hyperbolic groups. Essays in group theory, 75--263, Math. Sci. Res. Inst. Publ., 8, Springer, New York, 1987. • Bridson, Martin R.; Haefliger, André (1999). Metric spaces of non-positive curvature. Grundlehren der Mathematischen Wissenschaften 319. Berlin: Springer-Verlag. xxii+643. ISBN 3-540-64324-9. MR1744486 • Igor Mineyev, Bounded cohomology characterizes hyperbolic groups., Quart. J. Math. Oxford Ser., 53(2002), 59-73.

Further reading • É. Ghys and P. de la Harpe (editors), Sur les groupes hyperboliques d'après Mikhael Gromov. Progress in Mathematics, 83. Birkhäuser Boston, Inc., Boston, MA, 1990. xii+285 pp. ISBN 0-8176-3508-4 • Michel Coornaert, Thomas Delzant, Athanase Papadopoulos, "Géométrie et théorie des groupes : les groupes hyperboliques de Gromov", Lecture Notes in Mathematics, vol. 1441, Springer-Verlag, Berlin, 1990, MR 92f:57003, ISBN 3-540-52977-2

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Automatic group In mathematics, an automatic group is a finitely generated group equipped with several finite-state automata. These automata can tell if a given word representation of a group element is in a "canonical form" and can tell if two elements given in canonical words differ by a generator. More precisely, let G be a group and A be a finite set of generators. Then an automatic structure of G with respect to A is a set of finite-state automata: • the word-acceptor, which accepts for every element of G at least one word in A representing it • multipliers, one for each

, which accept a pair (w1, w2), for words wi accepted by the

word-acceptor, precisely when in G. The property of being automatic does not depend on the set of generators. The concept of automatic groups generalizes naturally to automatic semigroups.

Properties • Automatic groups have word problem solvable in quadratic time. A given word can actually be put into canonical form in quadratic time.

Examples of automatic groups • • • • • •

Finite groups, to see this take the regular language to be the set of all words in the finite group. Negatively curved groups Euclidean groups All finitely generated Coxeter groups [1] Braid groups Geometrically finite groups

Examples of non-automatic groups • Baumslag-Solitar groups • Non-Euclidean nilpotent groups

Biautomatic groups A group is biautomatic if it has two multipler automata, for left and right multiplication by elements of the generating set respectively. A biautomatic group is clearly automatic.[2] Examples include: • A hyperbolic group.[3] • An Artin group of finite type.[3]

Automatic group

References [1] Brink and Howlett (1993), "A finiteness property and an automatic structure for Coxeter groups", Mathematische Annalen (Springer Berlin / Heidelberg), ISSN 0025-5831. [2] Birget, Jean-Camille (2000), Algorithmic problems in groups and semigroups, Trends in mathematics, Birkhäuser, p. 82, ISBN 0817641300 [3] Charney, Ruth (1992), "Artin groups of finite type are biautomatic", Mathematische Annalen 292: 671–683, doi:10.1007/BF01444642

• Epstein, David B. A.; Cannon, James W.; Holt, Derek F.; Levy, Silvio V. F.; Paterson, Michael S.; Thurston, William P. (1992), Word Processing in Groups, Boston, MA: Jones and Bartlett Publishers, ISBN 0-86720-244-0. • Chiswell, Ian (2008), A Course in Formal Languages, Automata and Groups, Springer, ISBN 978-1-84800-939-4.

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Discrete group Concepts in group theory category of groups subgroups, normal subgroups group homomorphisms, kernel, image, quotient direct product, direct sum semidirect product, wreath product Types of groups simple, finite, infinite discrete, continuous multiplicative, additive cyclic, abelian, dihedral nilpotent, solvable list of group theory topics glossary of group theory

In mathematics, a discrete group is a group G equipped with the discrete topology. With this topology G becomes a topological group. A discrete subgroup of a topological group G is a subgroup H whose relative topology is the discrete one. For example, the integers, Z, form a discrete subgroup of the reals, R (with the standard metric topology), but the rational numbers, Q, do not. Any group can be given the discrete topology. Since every map from a discrete space is continuous, the topological homomorphisms between discrete groups are exactly the group homomorphisms between the underlying groups. Hence, there is an isomorphism between the category of groups and the category of discrete groups. Discrete groups can therefore be identified with their underlying (non-topological) groups. With this in mind, the term discrete group theory is used to refer to the study of groups without topological structure, in contradistinction to topological or Lie group theory. It is divided, logically but also technically, into finite group theory, and infinite group theory. There are some occasions when a topological group or Lie group is usefully endowed with the discrete topology, 'against nature'. This happens for example in the theory of the Bohr compactification, and in group cohomology theory of Lie groups.

Properties Since topological groups are homogeneous, one need only look at a single point to determine if the topological group is discrete. In particular, a topological group is discrete if and only if the singleton containing the identity is an open set. A discrete group is the same thing as a zero-dimensional Lie group (uncountable discrete groups are not second-countable so authors who require Lie groups to satisfy this axiom do not regard these groups as Lie groups). The identity component of a discrete group is just the trivial subgroup while the group of components is isomorphic to the group itself. Since the only Hausdorff topology on a finite set is the discrete one, a finite Hausdorff topological group must necessarily be discrete. It follows that every finite subgroup of a Hausdorff group is discrete. A discrete subgroup H of G is cocompact if there is a compact subset K of G such that HK = G.

Discrete group Discrete normal subgroups play an important role in the theory of covering groups and locally isomorphic groups. A discrete normal subgroup of a connected group G necessarily lies in the center of G and is therefore abelian. Other properties: • • • • • • • •

every discrete group is totally disconnected every subgroup of a discrete group is discrete. every quotient of a discrete group is discrete. the product of a finite number of discrete groups is discrete. a discrete group is compact if and only if it is finite. every discrete group is locally compact. every discrete subgroup of a Hausdorff group is closed. every discrete subgroup of a compact Hausdorff group is finite.

Examples • Frieze groups and wallpaper groups are discrete subgroups of the isometry group of the Euclidean plane. Wallpaper groups are cocompact, but Frieze groups are not. • A space group is a discrete subgroup of the isometry group of Euclidean space of some dimension. • A crystallographic group usually means a cocompact, discrete subgroup of the isometries of some Euclidean space. Sometimes, however, a crystallographic group can be a cocompact discrete subgroup of a nilpotent or solvable Lie group. • Every triangle group T is a discrete subgroup of the isometry group of the sphere (when T is finite), the Euclidean plane (when T has a Z + Z subgroup of finite index), or the hyperbolic plane. • Fuchsian groups are, by definition, discrete subgroups of the isometry group of the hyperbolic plane. • A Fuchsian group that preserves orientation and acts on the upper half-plane model of the hyperbolic plane is a discrete subgroup of the Lie group PSL(2,R), the group of orientation preserving isometries of the upper half-plane model of the hyperbolic plane. • A Fuchsian group is sometimes considered as a special case of a Kleinian group, by embedding the hyperbolic plane isometrically into three dimensional hyperbolic space and extending the group action on the plane to the whole space. • The modular group is PSL(2,Z), thought of as a discrete subgroup of PSL(2,R). The modular group is a lattice in PSL(2,R), but it is not cocompact. • Kleinian groups are, by definition, discrete subgroups of the isometry group of hyperbolic 3-space. These include quasi-Fuchsian groups. • A Kleinian group that preserves orientation and acts on the upper half space model of hyperbolic 3-space is a discrete subgroup of the Lie group PSL(2,C), the group of orientation preserving isometries of the upper half-space model of hyperbolic 3-space. • A lattice in a Lie group is a discrete subgroup such that the Haar measure of the quotient space is finite.

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References • Hazewinkel, Michiel, ed. (2001), "Discrete group of transformations" [1], Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104 • Hazewinkel, Michiel, ed. (2001), "Discrete subgroup" [2], Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104

References [1] http:/ / eom. springer. de/ d/ d033080. htm [2] http:/ / eom. springer. de/ d/ d033150. htm

Todd–Coxeter algorithm In group theory, the Todd–Coxeter algorithm, discovered by J.A. Todd and H.S.M. Coxeter in 1936, is an algorithm for solving the coset enumeration problem. Given a presentation of a group G by generators and relations and a subgroup H of G, the algorithm enumerates the cosets of H on G and describes the permutation representation of G on the space of the cosets. If the order of a group G is relatively small and the subgroup H is known to be uncomplicated (for example, a cyclic group), then the algorithm can be carried out by hand and gives a reasonable description of the group G. Using their algorithm, Coxeter and Todd showed that certain systems of relations between generators of known groups are complete, i.e. constitute systems of defining relations. The Todd–Coxeter algorithm can be applied to infinite groups and is known to terminate in a finite number of steps, provided that the index of H in G is finite. On the other hand, for a general pair consisting of a group presentation and a subgroup, its running time is not bounded by any computable function of the index of the subgroup and the size of the input data.

Description of the algorithm One implementation of the algorithm proceeds as follows. Suppose that generators and

is a set of relations and denote by where the

, where

the set of generators

are words of elements of

used: a coset table, a relation table for each relation in

is a set of

and their inverses. Let

. There are three types of tables that will be

, and a subgroup table for each generator

of

.

Information is gradually added to these tables, and once they are filled in, all cosets have been enumerated and the algorithm terminates. The coset table is used to store the relationships between the known cosets when multiplying by a generator. It has rows representing cosets of and a column for each element of . Let denote the coset of the ith row of the coset table, and let

denote generator of the jth column. The entry of the coset table in row i, column j is

defined to be (if known) k, where k is such that . The relation tables are used to detect when some of the cosets we have found are actually equivalent. One relation table for each relation in is maintained. Let be a relation in , where . The relation table has rows representing the cosets of

, as in the coset table. It has t columns, and the entry in the ith

row and jth column is defined to be (if known) k, where

. In particular, the

'th entry

is initially i, since . Finally, the subgroup tables are similar to the relation tables, except that they keep track of possible relations of the generators of . For each generator of , with , we create a subgroup table. It has only one row, corresponding to the coset of defined (if known) to be k, where

itself. It has t columns, and the entry in the jth column is .

Todd–Coxeter algorithm When a row of a relation or subgroup table is completed, a new piece of information

106 ,

, is

found. This is known as a deduction. From the deduction, we may be able to fill in additional entries of the relation and subgroup tables, resulting in possible additional deductions. We can fill in the entries of the coset table corresponding to the equations and . However, when filling in the coset table, it is possible that we may already have an entry for the equation, but the entry has a different value. In this case, we have discovered that two of our cosets are actually the same, known as a coincidence. Suppose , with . We replace all instances of j in the tables with i. Then, we fill in all possible entries of the tables, possibly leading to more deductions and coincidences. If there are empty entries in the table after all deductions and coincidences have been taken care of, add a new coset to the tables and repeat the process. We make sure that when adding cosets, if Hx is a known coset, then Hxg will be added at some point for all . (This is needed to guarantee that the algorithm will terminate provided is finite.) When all the tables are filled, the algorithm terminates. We then have all needed information on the action of the cosets of

.

See also • Coxeter group

References • J.A. Todd, H.S.M. Coxeter, A practical method for enumerating cosets of a finite abstract group. Proc. Edinb. Math. Soc., II. Ser. 5, 26-34 (1936). Zbl: 0015.10103, JFM 62.1094.02 • H.S.M. Coxeter, W.O.J. Moser, Generators and relations for discrete groups. Fourth edition. Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], 14. Springer-Verlag, Berlin-New York, 1980. ix+169 pp. ISBN 3-540-09212-9 MR0562913 • Seress, A. "An Introduction to Computational Group Theory" Notices of the AMS, June/July 1997.

on

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Frobenius group In mathematics, a Frobenius group is a transitive permutation group on a finite set, such that no non-trivial element fixes more than one point and some non-trivial element fixes a point. They are named after F. G. Frobenius.

Structure The subgroup H of a Frobenius group G fixing a point of the set X is called the Frobenius complement. The identity element together with all elements not in any conjugate of H form a normal subgroup called the Frobenius kernel K. (This is a theorem due to Frobenius.) The Frobenius group G is the semidirect product of K and H: . Both the Frobenius kernel and the Frobenius complement have very restricted structures. J. G. Thompson (1960) proved that the Frobenius kernel K is a nilpotent group. If H has even order then K is abelian. The Frobenius complement H has the property that every subgroup whose order is the product of 2 primes is cyclic; this implies that its Sylow subgroups are cyclic or generalized quaternion groups. Any group such that all Sylow subgroups are cyclic is called a Z-group, and in particular must be a metacyclic group: this means it is the extension of two cyclic groups. If a Frobenius complement H is not solvable then Zassenhaus showed that it has a normal subgroup of index 1 or 2 that is the product of SL2(5) and a metacyclic group of order coprime to 30. In particular, if a Frobenius complement coincides with its derived subgroup, then it is isomorphic with SL(2,5). If a Frobenius complement H is solvable then it has a normal metacyclic subgroup such that the quotient is a subgroup of the symmetric group on 4 points. A finite group is a Frobenius complement if and only if it has a faithful, finite-dimensional representation over a finite field in which non-identity group elements correspond to linear transformations without nonzero fixed points. The Frobenius kernel K is uniquely determined by G as it is the Fitting subgroup, and the Frobenius complement is uniquely determined up to conjugacy by the Schur-Zassenhaus theorem. In particular a finite group G is a Frobenius group in at most one way.

Examples • The smallest example is the symmetric group on 3 points, with 6 elements. The Frobenius kernel K has order 3, and the complement H has order 2.

The Fano plane

• For every finite field Fq with q (> 2) elements, the group of invertible affine transformations

,

acting naturally on Fq is a Frobenius group. The preceding example corresponds to the case F3, the field with three elements. • Another example is provided by the subgroup of order 21 of the collineation group of the Fano plane generated by a 3-fold symmetry σ fixing a point and a cyclic permutation τ of all 7 points, satisfying στ =τ²σ. Identifying F8* with the Fano plane, σ can be taken to be the restriction of the Frobenius automorphism σ(x)=x² of F8 and τ to be multiplication by any element not in the prime field F2 (i.e. a generator of the cyclic multiplicative group of F8). This Frobenius group acts simply transitively on the 21 flags in the Fano plane, i.e. lines with marked points.

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• The dihedral group of order 2n with n odd is a Frobenius group with complement of order 2. More generally if K is any abelian group of odd order and H has order 2 and acts on K by inversion, then the semidirect product K.H is a Frobenius group. • Many further examples can be generated by the following constructions. If we replace the Frobenius complement of a Frobenius group by a non-trivial subgroup we get another Frobenius group. If we have two Frobenius groups K1.H and K2.H then (K1 × K2).H is also a Frobenius group. • If K is the non-abelian group of order 73 with exponent 7, and H is the cyclic group of order 3, then there is a Frobenius group G that is an extension K.H of H by K. This gives an example of a Frobenius group with non-abelian kernel. This was the first example of Frobenius group with nonabelian kernel (it was constructed by Otto Schmidt). • If H is the group SL2(F5) of order 120, it acts fixed point freely on a 2-dimensional vector space K over the field with 11 elements. The extension K.H is the smallest example of a non-solvable Frobenius group. • The subgroup of a Zassenhaus group fixing a point is a Frobenius group. • Frobenius groups whose Fitting subgroup has arbitrarily large nilpotency class were constructed by Ito: Let q be a prime power, d a positive integer, and p a prime divisor of q −1 with d ≤ p. Fix some field F of order q and some element z of this field of order p. The Frobenius complement H is the cyclic subgroup generated by the diagonal matrix whose i,i'th entry is zi. The Frobenius kernel K is the Sylow q-subgroup of GL(d,q) consisting of upper triangular matrices with ones on the diagonal. The kernel K has nilpotency class d −1, and the semidirect product KH is a Frobenius group.

Representation theory The irreducible complex representations of a Frobenius group G can be read off from those of H and K. There are two types of irreducible representations of G: • Any irreducible representation R of H gives an irreducible representation of G using the quotient map from G to H (that is, as a restricted representation). These give the irreducible representations of G with K in their kernel. • If S is any non-trivial irreducible representation of K, then the corresponding induced representation of G is also irreducible. These give the irreducible representations of G with K not in their kernel.

Alternative definitions There are a number of group theoretical properties which are interesting on their own right, but which happen to be equivalent to the group possessing a permutation representation that makes it a Frobenius group. • G is a Frobenius group if and only if G has a proper, nonidentity subgroup H such that H ∩ Hg is the identity subgroup for every g ∈ G − H. This definition is then generalized to the study of trivial intersection sets which allowed the results on Frobenius groups used in the classification of CA groups to be extended to the results on CN groups and finally the odd order theorem. Assuming that

is the semidirect product of the normal subgroup K and complement H, then the

following restrictions on centralizers are equivalent to G being a Frobenius group with Frobenius complement H: • The centralizer CG(k) is a subgroup of K for every nonidentity k in K. • CH(k) = 1 for every nonidentity k in K. • CG(h) ≤ H for every nonidentity h in H.

Frobenius group

References • • • •

B. Huppert, Endliche Gruppen I, Springer 1967 I. M. Isaacs, Character theory of finite groups, AMS Chelsea 1976 D. S. Passman, Permutation groups, Benjamin 1968 Thompson, John G. (1960), "Normal p-complements for finite groups", Mathematische Zeitschrift 72: 332–354, doi:10.1007/BF01162958, MR0117289, ISSN 0025-5874

Zassenhaus group In mathematics, a Zassenhaus group, named after Hans Julius Zassenhaus, is a certain sort of doubly transitive permutation group very closely related to rank-1 groups of Lie type.

Definition A Zassenhaus group is a permutation group G on a finite set X with the following three properties: • G is doubly transitive. • Non-trivial elements of G fix at most two points. • G has no regular normal subgroup. ("Regular" means that non-trivial elements do not fix any points of X; compare free action.) The degree of a Zassenhaus group is the number of elements of X. Some authors omit the third condition that G has no regular normal subgroup. This condition is put in to eliminate some "degenerate" cases. The extra examples one gets by omitting it are either Frobenius groups or certain groups of degree 2p and order 2p(2p − 1)p for a prime p, that are generated by all semilinear mappings and Galois automorphisms of a field of order 2p.

Examples We let q = pf be a power of a prime p, and write Fq for the finite field of order q. Suzuki proved that any Zassenhaus group is of one of the following four types: • The projective special linear group PSL2(Fq) for q > 3 odd, acting on the q + 1 points of the projective line. It has order (q + 1)q(q − 1)/2. • The projective general linear group PGL2(Fq) for q > 3. It has order (q + 1)q(q − 1). • A certain group containing PSL2(Fq) with index 2, for q an odd square. It has order (q + 1)q(q − 1). • The Suzuki group Suz(Fq) for q a power of 2 that is at least 8 and not a square. The order is (q2 + 1)q2(q − 1) The degree of these groups is q + 1 in the first three cases, q2 + 1 in the last case.

Further reading • Finite Groups III (Grundlehren Der Mathematischen Wissenschaften Series, Vol 243) by B. Huppert, N. Blackburn, ISBN 0-387-10633-2

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Regular ''p''-group

Regular p-group In mathematical finite group theory, the concept of regular p-group captures some of the more important properties of abelian p-groups, but is general enough to include most "small" p-groups. Regular p-groups were introduced by Phillip Hall (1933).

Definition A finite p-group G is said to be regular if any of the following equivalent (Hall 1959, Ch. 12.4), (Huppert 1967, Kap. III §10) conditions are satisfied: • For every a, b in G, there is a c in the derived subgroup H′ of the subgroup H of G generated by a and b, such that ap · bp = (ab)p · cp. • For every a, b in G, there are elements ci in the derived subgroup of the subgroup generated by a and b, such that ap · bp = (ab)p · c1p ⋯ ckp. • For every a, b in G and every positive integer n, there are elements ci in the derived subgroup of the subgroup generated by a and b such that aq · bq = (ab)q · c1q ⋯ ckq, where q = pn.

Examples Many familiar p-groups are regular: • • • •

Every abelian p-group is regular. Every p-group of nilpotency class strictly less than p is regular. Every p-group of order at most pp is regular. Every finite group of exponent p is regular.

However, many familiar p-groups are not regular: • Every nonabelian 2-group is irregular. • The Sylow p-subgroup of the symmetric group on p2 points is irregular and of order pp+1.

Properties A p-group is regular if and only if every subgroup generated by two elements is regular. Every subgroup and quotient group of a regular group is regular, but the direct product of regular groups need not be regular. A 2-group is regular if and only if it is abelian. A 3-group with two generators is regular if and only if its derived subgroup is cyclic. Every p-group of odd order with cyclic derived subgroup is regular. The subgroup of a p-group G generated by the elements of order dividing pk is denoted Ωk(G) and regular groups are well-behaved in that Ωk(G) is precisely the set of elements of order dividing pk. The subgroup generated by all pk-th powers of elements in G is denoted ℧k(G). In a regular group, the index [G:℧k(G)] is equal to the order of Ωk(G). In fact, commutators and powers interact in particularly simple ways (Huppert 1967, Kap III §10, Satz 10.8). For example, given normal subgroups M and N of a regular p-group G and nonnegative integers m and n, one has [℧m(M),℧n(N)] = ℧m+n([M,N]). • Philip Hall's criteria of regularity of a p-group G: G is regular, if one of the following hold: 1. [G:℧1(G)] < pp 2. [G′:℧1(G′)| < pp−1 3. |Ω1(G)| < pp−1

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Generalizations • Powerful p-group • power closed p-group

References • Hall, Marshall (1959), The theory of groups, Macmillan, MR0103215 • Hall, Philip (1933), "A contribution to the theory of groups of prime-power order", Proceedings of the London Mathematical Society, second series 36: 29–95, doi:10.1112/plms/s2-36.1.29 • Huppert, B. (1967) (in German), Endliche Gruppen, Berlin, New York: Springer-Verlag, pp. 90–93, MR0224703, ISBN 978-3-540-03825-2, OCLC 527050

Isoclinism of groups In mathematics, specifically group theory, isoclinism is an equivalence relation on groups that is broader than isomorphism, that is, any two groups that are isomorphic are isoclinic, but two isoclinic groups may not be isomorphic. The concept of isoclinism was introduced by Hall (1940) to help classify and understand p-groups, although applicable to all groups. Isoclinism remains an important part of the study of p-groups, and for instance §29 of Berkovich (2008) and §21.2 of Blackburn, Neumann & Venkataraman (2007) are devoted to it. Isoclinism also has vital consequences for the Schur multiplier and the associated aspects of character theory, as described in Suzuki (1982, p. 256) and Conway et al. (1985, Ch. 6.7).

Definition According to Struik (1960), two groups G and G' are isoclinic if the following three conditions hold: (1) G mod Z is isomorphic to G' mod Z', where Z is the center of G and Z' is the center of G', (2) the commutator subgroup of G is isomorphic to the commutator subgroup of G', and (3) "the isomorphisms of (1) and (2) can be selected in such a way that whenever aZ and bZ correspond respectively to a'Z' and b'Z' under 1), then (a, b) = a−1b−1ab corresponds to (a',b') under 2)."

Examples All Abelian groups are isoclinic since they are equal to their centers and their commutator subgroups are always the identity subgroup. Indeed, a group is isoclinic to an abelian group if and only if it is itself abelian, and G is isoclinic with G×A if and only if A is abelian. The dihedral, quasidihedral, and quaternion groups of order 2n are isoclinic for n≥3, Berkovich (2008, p. 285). Isoclinism divides p-groups into families, and the smallest members of each family are called stem groups. A group is a stem group if and only if Z(G) ≤ [G,G], that is, if and only if every element of the center of the group is contained in the derived subgroup (also called the commutator subgroup), Berkovich (2008, p. 287). Some enumeration results on isoclinism families are given in Blackburn, Neumann & Venkataraman (2007, p. 226). Another textbook treatment of isoclinism is given in Suzuki (1986, pp. 92–95), which describes in more detail the isomorphisms induced by an isoclinism. Isoclinism is important in theory of projective representations of finite groups, as all Schur covering groups of a group are isoclinic, a fact already hinted at by Hall according to Suzuki (1982, p. 256). This is important in describing the character tables of the finite simple groups, and so is described in some detail in Conway et al. (1985, Ch. 6.7).

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References • Berkovich, Yakov (2008), Groups of prime power order. Vol. 1, de Gruyter Expositions in Mathematics, 46, Walter de Gruyter GmbH & Co. KG, Berlin, doi:10.1515/9783110208221.285, MR2464640, ISBN 978-3-11-020418-6 • Blackburn, Simon R.; Neumann, Peter M.; Venkataraman, Geetha (2007) (in English), Enumeration of finite groups, Cambridge Tracts in Mathematics no 173 (1st ed.), Cambridge University Press, ISBN 978-0-521-88217-0, OCLC 154682311 • Conway, John Horton; Curtis, R. T.; Norton, S. P.; Parker, R. A.; Wilson, R. A. (1985), Atlas of finite groups, Oxford University Press, MR827219, ISBN 978-0-19-853199-9 • Hall, Philip (1940), "The classification of prime-power groups" [1], Journal für die reine und angewandte Mathematik 182: 130–141, doi:10.1515/crll.1940.182.130, MR0003389, ISSN 0075-4102 • Struik, Ruth Rebekka (1960), "A note on prime-power groups" [2], Canadian Mathematical Bulletin 3: 27–30, MR0148744, ISSN 0008-4395 • Suzuki, Michio (1982), Group theory. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 247, Berlin, New York: Springer-Verlag, MR648772, ISBN 978-3-540-10915-0 • Suzuki, Michio (1986), Group theory. II, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 248, Berlin, New York: Springer-Verlag, MR815926, ISBN 978-0-387-10916-9

References [1] http:/ / resolver. sub. uni-goettingen. de/ purl?GDZPPN00217491X [2] http:/ / math. ca/ cmb/ v3/ p27

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Variety (universal algebra)

Variety (universal algebra) In mathematics, specifically universal algebra, a variety of algebras is the class of all algebraic structures of a given signature satisfying a given set of identities. Equivalently, a variety is a class of algebraic structures of the same signature which is closed under the taking of homomorphic images, subalgebras and (direct) products. In the context of category theory, a variety of algebras is usually called a finitary algebraic category. A covariety is the class of all coalgebraic structures of a given signature. A variety of algebras should not be confused with an algebraic variety. Intuitively, a variety of algebras is an equationally defined collection of algebras, while an algebraic variety is an equationally defined collection of elements from a single algebra. The two are named alike by analogy, but they are formally quite distinct and their theories have little in common.

Birkhoff's theorem Garrett Birkhoff proved equivalent the two definitions of variety given above, a result of fundamental importance to universal algebra and known as Birkhoff's theorem or as the HSP theorem. H, S, and P stand, respectively, for the closure operations of homomorphism, subalgebra, and product. An equational class for some signature Σ is the collection of all models, in the sense of model theory, that satisfy some set E of equations, asserting equality between terms. A model satisfies these equations if they are true in the model for any valuation of the variables. The equations in E are then said to be identities of the model. Examples of such identities are the commutative law, characterizing commutative algebras, and the absorption law, characterizing lattices. It is simple to see that the class of algebras satisfying some set of equations will be closed under the HSP operations. Proving the converse —classes of algebras closed under the HSP operations must be equational— is much harder.

Examples The class of all semigroups forms a variety of algebras of signature (2). A sufficient defining equation is the associative law:

It satisfies the HSP closure requirement, since any homomorphic image, any subset closed under multiplication and any direct product of semigroups is also a semigroup. The class of groups forms a class of algebras of signature (2,1,0), the three operations being respectively multiplication, inversion and identity. Any subset of a group closed under multiplication, under inversion and under identity (i.e. containing the identity) forms a subgroup. Likewise, the collection of groups is closed under homomorphic image and under direct product. Applying Birkhoff's theorem, this is sufficient to tell us that the groups form a variety, and so it should be defined by a collection of identities. In fact, the familiar axioms of associativity, inverse and identity form one suitable set of identities:

A subvariety is a subclass of a variety, closed under the operations H, S, P. Notice that although every group is a semigroup, the class of groups does not form a subvariety of the variety of semigroups. This is because not every subsemigroup of a group is a group.

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The class of abelian groups, considered again with signature (2,1,0), also has the HSP closure properties. It forms a subvariety of the variety of groups, and can be defined equationally by the three group axioms above together with the commutativity law:

Variety of finite algebras Since varieties are closed under arbitrary cartesian products, all non-trivial varieties contain infinite algebras. It follows that the theory of varieties is of limited use in the study of finite algebras, where one must often apply techniques particular to the finite case. With this in mind, attempts have been made to develop a finitary analogue of the theory of varieties. A variety of finite algebras, sometimes called a pseudovariety, is usually defined to be a class of finite algebras of a given signature, closed under the taking of homomorphic images, subalgebras and finitary direct products. There is no general finitary counterpart to Birkhoff's theorem, but in many cases the introduction of a more complex notion of equations allows similar results to be derived. Pseudovarieties are of particular importance in the study of finite semigroups and hence in formal language theory. Eilenberg's theorem, often referred to as the variety theorem describes a natural correspondence between varieties of regular languages and pseudovarieties of finite semigroups.

Category theory If A is a finitary algebraic category, then the forgetful functor

is monadic. Even more, it is strictly monadic, in that the comparison functor is an isomorphism (and not just an equivalence).[1] Here,

is the Eilenberg–Moore category on

general, one says a category is an algebraic category if it is monadic over

. In

. This is a more general notion than

"finitary algebraic category" (the notion of "variety" used in universal algebra) because it admits such categories as CABA (complete atomic Boolean algebras) and CSLat (complete semilattices) whose signatures include infinitary operations. In those two cases the signature is large, meaning that it forms not a set but a proper class, because its operations are of unbounded arity. The algebraic category of sigma algebras also has infinitary operations, but their arity is countable whence its signature is small (forms a set).

See also • Quasivariety

Notes [1] Saunders Mac Lane, Categories for the Working Mathematician, Springer. (See p. 152)

References Two monographs available free online: • Burris, Stanley N., and H.P. Sankappanavar, H. P., 1981. A Course in Universal Algebra. (http://www.thoralf. uwaterloo.ca/htdocs/ualg.html) Springer-Verlag. ISBN 3-540-90578-2. • Jipsen, Peter, and Henry Rose, 1992. Varieties of Lattices (http://www1.chapman.edu/~jipsen/ JipsenRoseVoL.html), Lecture Notes in Mathematics 1533. Springer Verlag. ISBN 0-387-56314-8.

Reflection group

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Reflection group In group theory and geometry, a reflection group is a discrete group which is generated by a set of reflections of a finite-dimensional Euclidean space. The symmetry group of a regular polytope or of a tiling of the Euclidean space by congruent copies of a regular polytope is necessarily a reflection group. Reflection groups also include Weyl groups and crystallographic Coxeter groups. While the orthogonal group is generated by reflections (by the Cartan–Dieudonné theorem), it is a continuous group (indeed, Lie group), not a discrete group, and is generally considered separately.

Definition Let E be a finite-dimensional Euclidean space. A finite reflection group is a subgroup of the general linear group of E which is generated by a set of orthogonal reflections across hyperplanes passing through the origin. An affine reflection group is a discrete subgroup of the affine group of E that is generated by a set of affine reflections of E (without the requirement that the reflection hyperplanes pass through the origin). The corresponding notions can be defined over other fields, leading to complex reflection groups and analogues of reflection groups over a finite field.

Examples Plane In two dimensions, the finite reflection groups are the dihedral groups, which are generated by reflection in two lines that form an angle of and correspond to the Coxeter diagram Conversely, the cyclic point groups in two dimensions are not generated by reflections, and indeed contain no reflections – they are however subgroups of index 2 of a dihedral group. Infinite reflection groups include the frieze groups

and

and the wallpaper groups pmm, p3m1, p4m,

and p6m. If the angle between two lines is an irrational multiple of pi, the group generated by reflections in these lines is infinite and non-discrete, hence, it is not a reflection group.

Space Finite reflection groups are the point groups Cnv, Dnh, and the symmetry groups of the five Platonic solids. Dual regular polyhedra (cube and octahedron, as well as dodecahedron and icosahedron) give rise to isomorphic symmetry groups. The classification of finite reflection groups of R3 is an instance of the ADE classification.

Kaleidoscopes Reflection groups have deep relations with kaleidoscopes, as discussed in (Goodman 2004).

Relation with Coxeter groups A reflection group W admits a presentation of a special kind discovered and studied by H.S.M. Coxeter. The reflections in the faces of a fixed fundamental "chamber" are generators ri of W of order 2. All relations between them formally follow from the relations

expressing the fact that the product of the reflections ri and rj in two hyperplanes Hi and Hj meeting at an angle is a rotation by the angle fixing the subspace Hi ∩ Hj of codimension 2. Thus, viewed as an abstract group, every reflection group is a Coxeter group.

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Finite fields When working over finite fields, one defines a "reflection" as a map that fixes a hyperplane (otherwise for example there would be no reflections in characteristic 2, as so reflections are the identity). Geometrically, this amounts to including shears in a hyperplane. Reflection groups over finite fields of characteristic not 2 were classified in (Zalesskiĭ & Serežkin 1981).

Generalizations Discrete isometry groups of more general Riemannian manifolds generated by reflections have also been considered. The most important class arises from Riemannian symmetric spaces of rank 1: the n-sphere Sn, corresponding to finite reflection groups, the Euclidean space Rn, corresponding to affine reflection groups, and the hyperbolic space Hn, where the corresponding groups are called hyperbolic reflection groups. In two dimensions, triangle groups include reflection groups of all three kinds.

See also • Hyperplane arrangement • Chevalley–Shephard–Todd theorem

References Standard references include (Humphreys 1992) and (Grove & Benson 1996). • Coxeter, H.S.M. (1934), "Discrete groups generated by reflections", Ann. of Math. 35: 588–621 • Coxeter, H.S.M. (1935), "The complete enumeration of finite groups of the form

", J.

London Math. Soc. 10: 21–25 • Goodman, Roe (April 2004), "The Mathematics of Mirrors and Kaleidoscopes" (http://www.math.rutgers.edu/ ~goodman/pub/monthly.pdf), American Mathematical Monthly • Humphreys, James E. (1992), Reflection groups and Coxeter groups, Cambridge University Press, ISBN 978-0-521-43613-7 • Zalesskiĭ, A E; Serežkin, V N (1981), "Finite Linear Groups Generated by Reflections", Math. USSR Izv. 17 (3): 477–503, doi:10.1070/IM1981v017n03ABEH001369 • Kane, Richard, Reflection groups and invariant theory (review) (http://www.cms.math.ca/Publications/ Reviews/2003/rev4.pdf) • Hartmann, Julia; Shepler, Anne V., Jacobians of reflection groups (http://arxiv.org/abs/math/0405135) • Dolgachev, Igor V., Reflection groups in algebraic geometry (http://arxiv.org/abs/math.AG/0610938)

External links • E. B. Vinberg (2001), "Reflection group" (http://eom.springer.de/R/r080520.htm), in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104

Fundamental group

Fundamental group In mathematics, more specifically algebraic topology, the fundamental group (discovered by Henri Poincaré who gave the definition in his article Analysis Situs, published in 1895) is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other. Intuitively, it records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest of the homotopy groups. Fundamental groups can be studied using the theory of covering spaces, since a fundamental group coincides with the group of deck transformations of the associated universal covering space. Its abelianisation can be identified with the first homology group of the space. When the topological space is homeomorphic to a simplicial complex, its fundamental group can be described explicitly in terms of generators and relations. Historically, the concept of fundamental group first emerged in the theory of Riemann surfaces, in the work of Bernhard Riemann, Henri Poincaré and Felix Klein, where it describes the monodromy properties of complex functions, as well as providing a complete topological classification of closed surfaces.

Intuition Start with a space (e.g. a surface), and some point in it, and all the loops both starting and ending at this point — paths that start at this point, wander around and eventually return to the starting point. Two loops can be combined together in an obvious way: travel along the first loop, then along the second. Two loops are considered equivalent if one can be deformed into the other without breaking. The set of all such loops with this method of combining and this equivalence between them is the fundamental group. For the precise definition, let X be a topological space, and let x0 be a point of X. We are interested in the set of continuous functions f : [0,1] → X with the property that f(0) = x0 = f(1). These functions are called loops with base point x0. Any two such loops, say f and g, are considered equivalent if there is a continuous function h : [0,1] × [0,1] → X with the property that, for all 0 ≤ t ≤ 1, h(t, 0) = f(t), h(t, 1) = g(t) and h(0, t) = x0 = h(1, t). Such an h is called a homotopy from f to g, and the corresponding equivalence classes are called homotopy classes. The product f ∗ g of two loops f and g is defined by setting (f ∗ g)(t) := f(2t) if 0 ≤ t ≤ 1/2 and (f ∗ g)(t) := g(2t − 1) if 1/2 ≤ t ≤ 1. Thus the loop f ∗ g first follows the loop f with "twice the speed" and then follows g with twice the speed. The product of two homotopy classes of loops [f] and [g] is then defined as [f ∗ g], and it can be shown that this product does not depend on the choice of representatives. With the above product, the set of all homotopy classes of loops with base point x0 forms the fundamental group of X at the point x0 and is denoted or simply π(X, x0). The identity element is the constant map at the basepoint, and the inverse of a loop f is the loop g defined by g(t) = f(1 − t). That is, g follows f backwards. Although the fundamental group in general depends on the choice of base point, it turns out that, up to isomorphism, this choice makes no difference so long as the space X is path-connected. For path-connected spaces, therefore, we can write π1(X) instead of π1(X, x0) without ambiguity whenever we care about the isomorphism class only.

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Examples Trivial fundamental group. In Euclidean space Rn, or any convex subset of Rn, there is only one homotopy class of loops, and the fundamental group is therefore the trivial group with one element. A path-connected space with a trivial fundamental group is said to be simply connected. Infinite cyclic fundamental group. The circle. Each homotopy class consists of all loops which wind around the circle a given number of times (which can be positive or negative, depending on the direction of winding). The product of a loop which winds around m times and another that winds around n times is a loop which winds around m + n times. So the fundamental group of the circle is isomorphic to , the additive group of integers. This fact can be used to give proofs of the Brouwer fixed point theorem and the Borsuk–Ulam theorem in dimension 2. Since the fundamental group is a homotopy invariant, the theory of the winding number for the complex plane minus one point is the same as for the circle. Free groups of higher rank: Graphs. Unlike the homology groups and higher homotopy groups associated to a topological space, the fundamental group need not be abelian. For example, the fundamental group of the figure eight is the free group on two letters. More generally, the fundamental group of any graph G is a free group. Here the rank of the free group is equal to 1 − χ(G): one minus the Euler characteristic of G, when G is connected. Knot theory. A somewhat more sophisticated example of a space with a non-abelian fundamental group is the complement of a trefoil knot in R3.

Functoriality If f : X → Y is a continuous map, x0 ∈ X and y0 ∈ Y with f(x0) = y0, then every loop in X with base point x0 can be composed with f to yield a loop in Y with base point y0. This operation is compatible with the homotopy equivalence relation and with composition of loops. The resulting group homomorphism, called the induced homomorphism, is written as π(f) or, more commonly,

We thus obtain a functor from the category of topological spaces with base point to the category of groups. It turns out that this functor cannot distinguish maps which are homotopic relative to the base point: if f and g : X → Y are continuous maps with f(x0) = g(x0) = y0, and f and g are homotopic relative to {x0}, then f* = g*. As a consequence, two homotopy equivalent path-connected spaces have isomorphic fundamental groups:

As an important special case, if X is path-connected then any two basepoints give isomorphic fundamental groups, with isomorphism given by a choice of path between the given basepoints. The fundamental group functor takes products to products and coproducts to coproducts. That is, if X and Y are path connected, then

and

(In the latter formula, denotes the wedge sum of topological spaces, and * the free product of groups.) Both formulas generalize to arbitrary products. Furthermore the latter formula is a special case of the Seifert–van Kampen theorem which states that the fundamental group functor takes pushouts along inclusions to pushouts.

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Fibrations A generalization of a product of spaces is given by a fibration,

Here the total space E is a sort of "twisted product" of the base space B and the fiber F. In general the fundamental groups of B, E and F are terms in a long exact sequence involving higher homotopy groups. When all the spaces are connected, this has the following consequences for the fundamental groups: • π1(B) and π1(E) are isomorphic if F is simply connected • πn+1(B) and πn(F) are isomorphic if E is contractible The latter is often applied to the situation E = path space of B, F = loop space of B or B = classifying space BG of a topological group G, E = universal G-bundle EG.

Relationship to first homology group The fundamental groups of a topological space X are related to its first singular homology group, because a loop is also a singular 1-cycle. Mapping the homotopy class of each loop at a base point x0 to the homology class of the loop gives a homomorphism from the fundamental group π1(X, x0) to the homology group H1(X). If X is path-connected, then this homomorphism is surjective and its kernel is the commutator subgroup of π1(X, x0), and H1(X) is therefore isomorphic to the abelianization of π1(X, x0). This is a special case of the Hurewicz theorem of algebraic topology.

Universal covering space If X is a topological space that is path connected, locally path connected and locally simply connected, then it has a simply connected universal covering space on which the fundamental group π(X,x0) acts freely by deck transformations with quotient space X. This space can be constructed analogously to the fundamental group by taking pairs (x, γ), where x is a point in X and γ is a homotopy class of paths from x0 to x and the action of π(X, x0) is by concatenation of paths. It is uniquely determined as a covering space.

Examples Let G be a connected, simply connected compact Lie group, for example the special unitary group SUn, and let Γ be a finite subgroup of G. Then the homogeneous space X = G/Γ has fundamental group Γ, which acts by right multiplication on the universal covering space G. Among the many variants of this construction, one of the most important is given by locally symmetric spaces X = Γ\G/K, where • G is a non-compact simply connected, connected Lie group (often semisimple), • K is a maximal compact subgroup of G • Γ is a discrete countable torsion-free subgroup of G. In this case the fundamental group is Γ and the universal covering space G/K is actually contractible (by the Cartan decomposition for Lie groups). As an example take G = SL2(R), K = SO2 and Γ any torsion-free congruence subgroup of the modular group SL2(Z). An even simpler example is given by G = R (so that K is trivial) and Γ = Z: in this case X=R/Z = S1. From the explicit realization, it also follows that the universal covering space of a path connected topological group H is again a path connected topological group G. Moreover the covering map is a continuous open homomorphism of G onto H with kernel Γ, a closed discrete normal subgroup of G:

Since G is a connected group with a continuous action by conjugation on a discrete group Γ, it must act trivially, so that Γ has to be a subgroup of the center of G. In particular π1(H) = Γ is an Abelian group; this can also easily be

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Fundamental group seen directly without using covering spaces. The group G is called the universal covering group of H. As the universal covering group suggests, there is an analogy between the fundamental group of a topological group and the center of a group; this is elaborated at Lattice of covering groups.

Edge-path group of a simplicial complex If X is a connected simplicial complex, an edge-path in X is defined to be a chain of vertices connected by edges in X. Two edge-paths are said to be edge-equivalent if one can be obtained from the other by successively switching between an edge and the two opposite edges of a triangle in X. If v is a fixed vertex in X, an edge-loop at v is an edge-path starting and ending at v. The edge-path group E(X, v) is defined to be the set of edge-equivalence classes of edge-loops at v, with product and inverse defined by concatenation and reversal of edge-loops. The edge-path group is naturally isomorphic to π1(|X|, v), the fundamental group of the geometric realisation |X| of X. Since it depends only on the 2-skeleton X2 of X (i.e. the vertices, edges and triangles of X), the groups π1(|X|,v) and π1(|X2|, v) are isomorphic. The edge-path group can be described explicitly in terms of generators and relations. If T is a maximal spanning tree in the 1-skeleton of X, then E(X, v) is canonically isomorphic to the group with generators the oriented edges of X not occurring in T and relations the edge-equivalences corresponding to triangles in X containing one or more edge not in T. A similar result holds if T is replaced by any simply connected—in particular contractible—subcomplex of X. This often gives a practical way of computing fundamental groups and can be used to show that every finitely presented group arises as the fundamental group of a finite simplicial complex. It is also one of the classical methods used for topological surfaces, which are classified by their fundamental groups. The universal covering space of a finite connected simplicial complex X can also be described directly as a simplicial complex using edge-paths. Its vertices are pairs (w,γ) where w is a vertex of X and γ is an edge-equivalence class of paths from v to w. The k-simplices containing (w,γ) correspond naturally to the k-simplices containing w. Each new vertex u of the k-simplex gives an edge wu and hence, by concatenation, a new path γu from v to u. The points (w,γ) and (u, γu) are the vertices of the "transported" simplex in the universal covering space. The edge-path group acts naturally by concatenation, preserving the simplicial structure, and the quotient space is just X. It is well-known that this method can also be used to compute the fundamental group of an arbitrary topological space. This was doubtless known to Čech and Leray and explicitly appeared as a remark in a paper by Weil (1960); various other authors such as L. Calabi, W-T. Wu and N. Berikashvili have also published proofs. In the simplest case of a compact space X with a finite open covering in which all non-empty finite intersections of open sets in the covering are contractible, the fundamental group can be identified with the edge-path group of the simplicial complex corresponding to the nerve of the covering.

Realizability • Every group can be realized as the fundamental group of a connected CW-complex of dimension 2 (or higher). As noted above, though, only free groups can occur as fundamental groups of 1-dimensional CW-complexes (that is, graphs). • Every finitely presented group can be realized as the fundamental group of a compact, connected, smooth manifold of dimension 4 (or higher). But there are severe restrictions on which groups occur as fundamental groups of low-dimensional manifolds. For example, no free abelian group of rank 4 or higher can be realized as the fundamental group of a manifold of dimension 3 or less.

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Fundamental group

Related concepts The fundamental group measures the 1-dimensional hole structure of a space. For studying "higher-dimensional holes", the homotopy groups are used. The elements of the n-th homotopy group of X are homotopy classes of (basepoint-preserving) maps from Sn to X. The set of loops at a particular base point can be studied without regarding homotopic loops as equivalent. This larger object is the loop space. For topological groups, a different group multiplication may be assigned to the set of loops in the space, with pointwise multiplication rather than concatenation. The resulting group is the loop group.

Fundamental groupoid Rather than singling out one point and considering the loops based at that point up to homotopy, one can also consider all paths in the space up to homotopy (fixing the initial and final point). This yields not a group but a groupoid, the fundamental groupoid of the space.

References • Joseph J. Rotman, An Introduction to Algebraic Topology, Springer-Verlag, ISBN 0-387-96678-1 • Isadore Singer and John A. Thorpe, Lecture Notes on Elementary Geometry and Topology, Springer-Verlag (1967) ISBN 0-387-90202-3 • Allen Hatcher, Algebraic Topology [1], Cambridge University Press (2002) ISBN 0-521-79540-0 • Peter Hilton and Shaun Wylie, Homology Theory, Cambridge University Press (1967) [warning: these authors use contrahomology for cohomology] • Richard Maunder, Algebraic Topology, Dover (1996) ISBN 0486691314 • Deane Montgomery and Leo Zippin, Topological Transformation Groups, Interscience Publishers (1955) • James Munkres, Topology, Prentice Hall (2000) ISBN 0131816292 • Herbert Seifert and William Threlfall, A Textbook of Topology (translated from German by Wofgang Heil), Academic Press (1980), ISBN 0126348502 • Edwin Spanier, Algebraic Topology, Springer-Verlag (1966) ISBN 0-387-94426-5 • André Weil, On discrete subgroups of Lie groups, Ann. of Math. 72 (1960), 369-384. • Fundamental group [2] on PlanetMath • Fundamental groupoid [3] on PlanetMath

Notes [1] http:/ / www. math. cornell. edu/ ~hatcher/ AT/ ATpage. html [2] http:/ / planetmath. org/ ?op=getobj& amp;from=objects& amp;id=849 [3] http:/ / planetmath. org/ ?op=getobj& amp;from=objects& amp;id=3941

External links • Dylan G.L. Allegretti, Simplicial Sets and van Kampen's Theorem (http://www.math.uchicago.edu/~may/ VIGRE/VIGREREU2008.html) (An elementary discussion of the fundamental groupoid of a topological space and the fundamental groupoid of a simplicial set). • Animations to introduce to the fundamental group by Nicolas Delanoue (http://www.istia.univ-angers.fr/ ~delanoue/topo_alg/)

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Classical group

Classical group In mathematics, the classical Lie groups are four infinite families of Lie groups closely related to the symmetries of Euclidean spaces. Their finite analogues are the classical groups of Lie type. The term was coined by Hermann Weyl (as seen in the title of his 1939 monograph The Classical Groups). Contrasting with the classical Lie groups are the exceptional Lie groups, which share their abstract properties, but not their familiarity. Sometimes classical groups are discussed in the restricted setting of compact groups, a formulation which makes their representation theory and algebraic topology easiest to handle. It does however exclude the general linear group.[1]

Relationship with bilinear forms The unifying feature of classical Lie groups is that they are close to the isometry groups of a certain bilinear or sesquilinear forms. The four series are labelled by the Dynkin diagram attached to it, with subscript n ≥ 1. The families may be represented as follows: • An = SU(n+1), the special unitary group of unitary n+1-by-n+1 complex matrices with determinant 1. • Bn = SO(2n+1), the special orthogonal group of orthogonal 2n+1-by-2n+1 real matrices with determinant 1. • Cn = Sp(n), the symplectic group of n-by-n quaternionic matrices that preserve the usual inner product on Hn. • Dn = SO(2n), the special orthogonal group of orthogonal 2n-by-2n real matrices with determinant 1. For certain purposes it is also natural to drop the condition that the determinant be 1 and consider unitary groups and (disconnected) orthogonal groups. The table lists the so-called connected compact real forms of the groups; they have closely-related complex analogues and various non-compact forms, for example, together with compact orthogonal groups one considers indefinite orthogonal groups. The Lie algebras corresponding to these groups are known as the classical Lie algebras. Viewing a classical group G as a subgroup of GL(n) via its definition as automorphisms of a vector space preserving some involution provides a representation of G called the standard representation.

Classical groups over general fields or rings Classical groups, more broadly considered in algebra, provide particularly interesting matrix groups. When the ring of coefficients of the matrix group is the real number or complex number field, these groups are just certain of the classical Lie groups. When the underlying ring is a finite field the classical groups are groups of Lie type. These groups play an important role in the classification of finite simple groups. Considering their abstract group theory, many linear groups have a "special" subgroup, usually consisting of the elements of determinant 1 (for orthogonal groups in characteristic 2 it consists of the elements of Dickson invariant 0), and most of them have associated "projective" quotients, which are the quotients by the center of the group. The word "general" in front of a group name usually means that the group is allowed to multiply some sort of form by a constant, rather than leaving it fixed. The subscript n usually indicates the dimension of the module on which the group is acting. Caveat: this notation clashes somewhat with the n of Dynkin diagrams, which is the rank.

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Classical group

General and special linear groups The general linear group GLn(R) is the group of all R-linear automorphisms of R^n. There is a subgroup: the special linear group SLn(R), and their quotients: the projective general linear group PGLn(R) = GLn(R)/Z(GLn(R)) and the projective special linear group PSLn(R) = SLn(R)/Z(SLn(R)). The projective special linear group PSLn(R) over a field R is simple for n≥2, except for the 2 cases when n=2 and the field has order 2 or 3.

Unitary groups The unitary group Un(R) is a group preserving a sesquilinear form on a module. There is a subgroup, the special unitary group SUn(R) and their quotients the projective unitary group PUn(R) = Un(R)/Z(Un(R)) and the projective special unitary group PSUn(R) = SUn(R)/Z(SUn(R))

Symplectic groups The symplectic group Sp2n(R) preserves a skew symmetric form on a module. It has a quotient, the projective symplectic group PSp2n(R). The general symplectic group GSp2n(R) consists of the automorphisms of a module multiplying a skew symmetric form by some invertible scalar. The projective symplectic group PSp2n(R) over a finite field R is simple for n≥1, except for the 2 cases when n=1 and the field has order 2 or 3.

Orthogonal groups The orthogonal group On(R) preserves a non-degenerate quadratic form on a module. There is a subgroup, the special orthogonal group SOn(R) and quotients, the projective orthogonal group POn(R), and the projective special orthogonal group PSOn(R). (In characteristic 2 the determinant is always 1, so the special orthogonal group is often defined as the subgroup of elements of Dickson invariant 1.) There is a nameless group often denoted by Ωn(R) consisting of the elements of the orthogonal group of elements of spinor norm 1, with corresponding subgroup and quotient groups SΩn(R), PΩn(R), PSΩn(R). (For positive definite quadratic forms over the reals, the group Ω happens to be the same as the orthogonal group, but in general it is smaller.) There is also a double cover of Ωn(R), called the pin group Pinn(R), and it has a subgroup called the spin group Spinn(R). The general orthogonal group GOn(R) consists of the automorphisms of a module multiplying a quadratic form by some invertible scalar.

Notes [1] Historically, in Klein's time, the most obvious example would have been the complex projective linear group, because it was the symmetry group of complex projective space, the dominant geometric concept of the nineteenth century. Vector spaces came later (indeed at the hands of Weyl, as an abstract algebraic notion), referring attention to their symmetry groups, the general linear groups. These groups are algebraic groups. In the development of the Langlands program, the general linear groups became central as the simplest and most universal cases.

References • E. Artin, Geometric algebra , Interscience (1957) • Dieudonné, Jean (1955), La géométrie des groupes classiques (http://books.google.com/ books?id=AfYZAQAAIAAJ), Ergebnisse der Mathematik und ihrer Grenzgebiete (N.F.), Heft 5, Berlin, New York: Springer-Verlag, MR0072144, ISBN 978-0-387-05391-2 • V. L. Popov (2001), "Classical group" (http://eom.springer.de/C/c022410.htm), in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104 • Weyl, The classical groups, ISBN 0691057567 • R.Slansky, Group theory for unified model building, Physics Reports, Volume 79, Issue 1, p. 1-128

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Unitary group

124

Unitary group In mathematics, the unitary group of degree n, denoted U(n), is the group of n×n unitary matrices, with the group operation that of matrix multiplication. The unitary group is a subgroup of the general linear group GL(n, C). Hyperorthogonal group is an archaic name for the unitary group, especially over finite fields. In the simple case n = 1, the group U(1) corresponds to the circle group, consisting of all complex numbers with absolute value 1 under multiplication. All the unitary groups contain copies of this group. The unitary group U(n) is a real Lie group of dimension n2. The Lie algebra of U(n) consists of complex n×n skew-Hermitian matrices, with the Lie bracket given by the commutator. The general unitary group (also called the group of unitary similitudes) consists of all matrices

such that

is a nonzero multiple of the identity matrix, and is just the product of the unitary group with the group of all positive multiples of the identity matrix.

Properties Since the determinant of a unitary matrix is a complex number with norm 1, the determinant gives a group homomorphism

The kernel of this homomorphism is the set of unitary matrices with unit determinant. This subgroup is called the special unitary group, denoted SU(n). We then have a short exact sequence of Lie groups:

This short exact sequence splits so that U(n) may be written as a semidirect product of SU(n) by U(1). Here the U(1) subgroup of U(n) consists of matrices of the form diag(eiθ, 1, 1, ..., 1). The unitary group U(n) is nonabelian for n > 1. The center of U(n) is the set of scalar matrices λI with λ ∈ U(1). This follows from Schur's lemma. The center is then isomorphic to U(1). Since the center of U(n) is a 1-dimensional abelian normal subgroup of U(n), the unitary group is not semisimple.

Topology The unitary group U(n) is endowed with the relative topology as a subset of Mn(C), the set of all n×n complex matrices, which is itself homeomorphic to a 2n2-dimensional Euclidean space. As a topological space, U(n) is both compact and connected. The compactness of U(n) follows from the Heine-Borel theorem and the fact that it is a closed and bounded subset of Mn(C). To show that U(n) is connected, recall that any unitary matrix A can be diagonalized by another unitary matrix S. Any diagonal unitary matrix must have complex numbers of absolute value 1 on the main diagonal. We can therefore write

A path in U(n) from the identity to A is then given by

The unitary group is not simply connected; the fundamental group of U(n) is infinite cyclic for all n:

The first unitary group U(1) is topologically a circle, which is well known to have a fundamental group isomorphic to Z, and the inclusion map is an isomorphism on . (It has quotient the Stiefel manifold.) The determinant map inducing the inverse.

induces an isomorphism of fundamental groups, with the splitting

Unitary group

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Related groups 2-out-of-3 property The unitary group is the 3-fold intersection of the orthogonal, symplectic, and complex groups:

Thus a unitary structure can be seen as an orthogonal structure, a complex structure, and a symplectic structure, which are required to be compatible (meaning that one uses the same J in the complex structure and the symplectic form, and that this J is orthogonal; writing all the groups as matrix groups fixes a J (which is orthogonal) and ensures compatibility). In fact, it is the intersection of any two of these three; thus a compatible orthogonal and complex structure induce a symplectic structure, and so forth. [1] [2] At the level of equations, this can be seen as follows: Symplectic: Complex: Orthogonal: Any two of these equations implies the third. At the level of forms, this can be seen by decomposing a Hermitian form into its real and imaginary parts: the real part is symmetric (orthogonal), and the imaginary part is skew-symmetric (symplectic)—and these are related by the complex structure (which is the compatibility). On an almost Kähler manifold, one can write this decomposition as , where h is the Hermitian form, g is the Riemannian metric, i is the almost complex structure, and is the almost symplectic structure. From the point of view of Lie groups, this can partly be explained as follows: subgroup of the intersection of so

, and

is the maximal compact subgroup of both or

is the maximal compact and

. Thus

is the maximal compact subgroup of both of these,

. From this perspective, what is unexpected is the intersection

.

Special unitary and projective unitary groups

Just as the orthogonal group has the special orthogonal group SO(n) as subgroup and the projective orthogonal group PO(n) as quotient, and the projective special orthogonal group PSO(n) as subquotient, the unitary group has associated to it the special unitary group SU(n), the projective unitary group PU(n), and the projective special unitary group PSU(n). These are related as by the commutative diagram at right; notably, both projective groups are equal: . The above is for the classical unitary group (over the complex numbers) – for unitary groups over finite fields, one similarly obtains special unitary and projective unitary groups, but in general .

Unitary group

126

G-structure: almost Hermitian In the language of G-structures, a manifold with a

-structure is an almost Hermitian manifold.

Generalizations From the point of view of Lie theory, the classical unitary group is a real form of the Steinberg group

, which is

an algebraic group that arises from the combination of the diagram automorphism of the general linear group (reversing the Dynkin diagram extension

, which corresponds to transpose inverse) and the field automorphism of the

(namely complex conjugation). Both these automorphisms are automorphisms of the algebraic

group, have order 2, and commute, and the unitary group is the fixed points of the product automorphism, as an algebraic group. The classical unitary group is a real form of this group, corresponding to the standard Hermitian form , which is positive definite. This can be generalized in a number of ways: • generalizing to other Hermitian forms yields indefinite unitary groups ; • the field extension can be replaced by any degree 2 separable algebra, most notably a degree 2 extension of a finite field; • generalizing to other diagrams yields other groups of Lie type, namely the other Steinberg groups (in addition to ) and Suzuki-Ree groups

• considering a generalized unitary group as an algebraic group, one can take its points over various algebras.

Indefinite forms Analogous to the indefinite orthogonal groups, one can define an indefinite unitary group, by considering the transforms that preserve a given Hermitian form, not necessarily positive definite (but generally taken to be non-degenerate). Here one is working with a vector space over the complex numbers. Given a Hermitian form

on a complex vector space

preserve the form: the transform

such that

, the unitary group

is the group of transforms that for all

. In terms of

matrices, representing the form by a matrix denoted , this says that . Just as for symmetric forms over the reals, Hermitian forms are determined by signature, and are all unitarily congruent to a diagonal form with entries of 1 on the diagonal and entries of . The non-degenerate assumption is equivalent to

. In a standard basis, this is represented as a quadratic form as:

and as a symmetric form as:

The resulting group is denoted

.

Finite fields Over the finite field with automorphism form on an

elements, (the

vector space

and

, there is a unique degree 2 extension field,

, with order 2

th power of the Frobenius automorphism). This allows one to define a Hermitian , as an

for

-bilinear map

such that

. Further, all non-degenerate Hermitian forms on a vector space over a

finite field are unitarily congruent to the standard one, represented by the identity matrix, that is, any Hermitian form is unitarily equivalent to

Unitary group

127

where

represent the coordinates of

in some particular

-basis of the

-dimensional space

(Grove 2002, Thm. 10.3). Thus one can define a (unique) unitary group of dimension or

for the extension

, denoted either as

depending on the author. The subgroup of the unitary group consisting of matrices of

determinant 1 is called the special unitary group and denoted article will use the

convention. The center of

matrices which are unitary, that is those matrices order

or

. For convenience, this

has order

with

and consists of the scalar

. The center of the special unitary group has

and consists of those unitary scalars which also have order dividing

unitary group by its center is called the projective unitary group,

, and the quotient of the special

unitary group by its center is the projective special unitary group ),

. The quotient of the

. In most cases (

is a perfect group and

and

is a finite simple

group, (Grove 2002, Thm. 11.22 and 11.26).

Degree-2 separable algebras

More generally, given a field k and a degree-2 separable k-algebra K (which may be a field extension but need not be), one can define unitary groups with respect to this extension. First, there is a unique k-automorphism of K only if

)

[3]

which is an involution and fixes exactly

(

if and

. This generalizes complex conjugation and the conjugation of degree 2 finite field extensions,

and allows one to define Hermitian forms and unitary groups as above.

Algebraic groups The equations defining a unitary group are polynomial equations over the equations are given in matrices as different form, they are -algebra

(but not over

, where

): for the standard form

is the conjugate transpose. Given a

. The unitary group is thus an algebraic group, whose points over a

are given by:

For the field extension

and the standard (positive definite) Hermitian form, these yield an algebraic group

with real and complex points given by:

Polynomial invariants The unitary groups are the automorphisms of two polynomials in real non-commutative variables:

These are easily seen to be the real and imginary parts of the complex form

. The two invariants separately are

invariants of O(2n) and Sp(2n,R). Combined they make the invariants of U(n) which is a subgroup of both these groups. The variables must be non-commutative in these invariants otherwise the second polynomial is identically zero.

Unitary group

Classifying space The classifying space for U(n) is described in the article classifying space for U(n).

See also • projective unitary group • orthogonal group • symplectic group

Notes [1] This is discussed in Arnold, "Mathematical Methods of Classical Mechanics". [2] symplectic (http:/ / www. math. ucr. edu/ home/ baez/ symplectic. html) [3] Milne, Algebraic Groups and Arithmetic Groups (http:/ / www. jmilne. org/ math/ CourseNotes/ aag. html), p. 103

References • Grove, Larry C. (2002), Classical groups and geometric algebra, Graduate Studies in Mathematics, 39, Providence, R.I.: American Mathematical Society, MR1859189, ISBN 978-0-8218-2019-3

Character theory This article refers to the use of the term character theory in mathematics, for the media studies definition see Character theory (Media). In mathematics, more specifically in group theory, the character of a group representation is a function on the group which associates to each group element the trace of the corresponding matrix. The character carries the essential information about the representation in a more condensed form. Georg Frobenius initially developed representation theory of finite groups entirely based on the characters, and without any explicit matrix realization of representations themselves. This is possible because a complex representation of a finite group is determined (up to isomorphism) by its character. The situation with representations over a field of positive characteristic, so-called "modular representations", is more delicate, but Richard Brauer developed a powerful theory of characters in this case as well. Many deep theorems on the structure of finite groups use characters of modular representations.

Applications Characters of irreducible representations encode many important properties of a group and can thus be used to study its structure. Character theory is an essential tool in the classification of finite simple groups. Close to half of the proof of the Feit–Thompson theorem involves intricate calculations with character values. Easier, but still essential, results that use character theory include the Burnside theorem (a purely group-theoretic proof of the Burnside theorem does exist, however), and a theorem of Richard Brauer and Michio Suzuki stating that a finite simple group cannot have a generalized quaternion group as its Sylow 2 subgroup.

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Character theory

129

Definitions Let V be a finite-dimensional vector space over a field F and let ρ:G → GL(V) be a representation of a group G on V. The character of ρ is the function χρ: G → F given by where

is the trace.

A character χρ is called irreducible if ρ is an irreducible representation. It is called linear if the dimension of ρ is 1. The kernel of a character χρ is the set: where χρ(1) is the value of χρ on the group identity. If ρ is a representation of G of dimension k and 1 is the identity of G then

Unlike the situation with the character group, the characters of a group do not, in general, form a group themselves.

Properties • Characters are class functions, that is, they each take a constant value on a given conjugacy class. • Isomorphic representations have the same characters. Over a field of characteristic 0, representations are isomorphic if and only if they have the same character. • If a representation is the direct sum of subrepresentations, then the corresponding character is the sum of the characters of those subrepresentations. • If a character of the finite group G is restricted to a subgroup H, then the result is also a character of H. • Every character value

is a sum of n mth roots of unity, where n is the degree (that is, the dimension of the

associated vector space) of the representation with character χ and m is the order of g. In particular, when F is the field of complex numbers, every such character value is an algebraic integer. • If F is the field of complex numbers, and

is irreducible, then

is an algebraic integer for

each x in G. • If F is algebraically closed and char(F) does not divide |G|, then the number of irreducible characters of G is equal to the number of conjugacy classes of G. Furthermore, in this case, the degrees of the irreducible characters are divisors of the order of G.

Arithmetic properties Let ρ and σ be representations of G. Then the following identities hold:

Character theory where

130 is the direct sum,

denotes the conjugate transpose of ρ, and Alt2 is

is the tensor product,

the alternating product Alt2 (ρ) =

and Sym2 is the symmetric square, which is determined by .

Character tables The irreducible complex characters of a finite group form a character table which encodes much useful information about the group G in a compact form. Each row is labelled by an irreducible character and the entries in the row are the values of that character on the representatives of the respective conjugacy class of G. The columns are labelled by (representatives of) the conjugacy classes of G. It is customary to label the first row by the trivial character, and the first column by (the conjugacy class of) the identity. The entries of the first column are the values of the irreducible characters at the identity, the degrees of the irreducible characters. Characters of degree 1 are known as linear characters. Here is the character table of

, the cyclic group with three elements and generator u: (1) (u) (u2) 1

1

1

1

χ1 1

ω

ω2

χ2 1

ω2 ω

where ω is a primitive third root of unity. The character table is always square, because the number of irreducible representations is equal to the number of conjugacy classes. The first row of the character table always consists of 1s, and corresponds to the trivial representation (the 1-dimensional representation consisting of 1×1 matrices containing the entry 1).

Orthogonality relations The space of complex-valued class functions of a finite group G has a natural inner-product:

where

means the complex conjugate of the value of

on g. With respect to this inner product, the irreducible

characters form an orthonormal basis for the space of class-functions, and this yields the orthogonality relation for the rows of the character table:

For

the orthogonality relation for columns is as follows:

where the sum is over all of the irreducible characters

of G and the symbol

denotes the order of the

centralizer of . The orthogonality relations can aid many computations including: • Decomposing an unknown character as a linear combination of irreducible characters. • Constructing the complete character table when only some of the irreducible characters are known. • Finding the orders of the centralizers of representatives of the conjugacy classes of a group.

Character theory

131

• Finding the order of the group.

Character table properties Certain properties of the group G can be deduced from its character table: • The order of G is given by the sum of the squares of the entries of the first column (the degrees of the irreducible characters). (See Representation theory of finite groups#Applying Schur's lemma.) More generally, the sum of the squares of the absolute values of the entries in any column gives the order of the centralizer of an element of the corresponding conjugacy class. • All normal subgroups of G (and thus whether or not G is simple) can be recognised from its character table. The kernel of a character χ is the set of elements g in G for which χ(g) = χ(1); this is a normal subgroup of G. Each normal subgroup of G is the intersection of the kernels of some of the irreducible characters of G. • The derived subgroup of G is the intersection of the kernels of the linear characters of G. In particular, G is Abelian if and only if all its irreducible characters are linear. • It follows, using some results of Richard Brauer from modular representation theory, that the prime divisors of the orders of the elements of each conjugacy class of a finite group can be deduced from its character table (an observation of Graham Higman). The character table does not in general determine the group up to isomorphism: for example, the quaternion group Q and the dihedral group of 8 elements (D4) have the same character table. Brauer asked whether the character table, together with the knowledge of how the powers of elements of its conjugacy classes are distributed, determines a finite group up to isomorphism. In 1964, this was answered in the negative by E. C. Dade. The linear characters form a character group, which has important number theoretic connections.

Induced characters and Frobenius reciprocity The characters discussed in this section are assumed to be complex-valued. Let H be a subgroup of the finite group G. Given a character of G, let denote its restriction to H. Let be a character of H. Ferdinand Georg Frobenius showed how to construct a character of G from

, using what is now known as Frobenius reciprocity.

Since the irreducible characters of G form an orthonormal basis for the space of complex-valued class functions of G, there is a unique class function of G with the property that

for each irreducible character

of G (the leftmost inner product is for class functions of G and the rightmost inner

product is for class functions of H). Since the restriction of a character of G to the subgroup H is again a character of H, this definition makes it clear that is a non-negative integer combination of irreducible characters of G, so is indeed a character of G. It is known as the character of G induced from θ. The defining formula of Frobenius reciprocity can be extended to general complex-valued class functions. Given a matrix representation ρ of H, Frobenius later gave an explicit way to construct a matrix representation of G, known as the representation induced from ρ, and written analogously as . This led to an alternative description of the induced character

. This induced character vanishes on all elements of G which are not conjugate to any

element of H. Since the induced character is a class function of G, it is only now necessary to describe its values on elements of H. Writing G as a disjoint union of right cosets of H, say and given an element h of H, the value

is precisely the sum of those

for which the conjugate

is also in H. Because θ is a class function of H, this value does not depend on the particular choice of coset representatives. This alternative description of the induced character sometimes allows explicit computation from relatively little information about the embedding of H in G, and is often useful for calculation of particular character tables. When θ

Character theory

132

is the trivial character of H, the induced character obtained is known as the permutation character of G (on the cosets of H). The general technique of character induction and later refinements found numerous applications in finite group theory and elsewhere in mathematics, in the hands of mathematicians such as Emil Artin, Richard Brauer, Walter Feit and Michio Suzuki, as well as Frobenius himself.

Mackey decomposition Mackey decomposition was defined and explored by George Mackey in the context of Lie groups, but is a powerful tool in the character theory and representation theory of finite groups. Its basic form concerns the way a character (or module) induced from a subgroup H of a finite group G behaves on restriction back to a (possibly different) subgroup K of G, and makes use of the decomposition of G into (H,K)-double cosets. If

is a disjoint union, and

where

is a complex class function of H, then Mackey's formula states that

is the class function of

defined by

for each h in H. There is a similar

formula for the restriction of an induced module to a subgroup, which holds for representations over any ring, and has applications in a wide variety of algebraic and topological contexts. Mackey decomposition, in conjunction with Frobenius reciprocity, yields a well-known and useful formula for the inner product of two class functions θ and ψ induced from respective subgroups H and K, whose utility lies in the fact that it only depends on how conjugates of H and K intersect each other. The formula (with its derivation) is:

(where T is a full set of (H,K)- double coset representatives, as before). This formula is often used when θ and ψ are linear characters, in which case all the inner products appearing in the right hand sum are either 1 or 0, depending on whether or not the linear characters θt and ψ have the same restriction to . If θ and ψ are both trivial characters, then the inner product simplifies to |T|.

"Twisted" dimension One may interpret the character of a representation as the "twisted" dimension of a vector space[1] – that is, a function parametrized by the group whose value on the identity is the dimension of the space, since Accordingly, one can view the other values

of the character as "twisted" dimensions, and find analogs or

generalizations of statements about dimensions to statements about characters or representations. A sophisticated example of this occurs in the theory of monstrous moonshine: the j-invariant is the graded dimension of an infinite-dimensional graded representation of the Monster group, and replacing the dimension with the character gives the McKay–Thompson series for each element of the Monster group.[1]

Character theory

References [1] (Gannon 2006)

• Lecture 2 of Fulton, William; Harris, Joe (1991), Representation theory. A first course, Graduate Texts in Mathematics, Readings in Mathematics, 129, New York: Springer-Verlag, MR1153249, ISBN 978-0-387-97527-6, ISBN 978-0-387-97495-8 • Isaacs, I.M. (1994). Character Theory of Finite Groups (Corrected reprint of the 1976 original, published by Academic Press. ed.). Dover. ISBN 0-486-68014-2. • Gannon, Terry (2006). Moonshine beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics. ISBN 0-521-83531-3 • James, Gordon; Liebeck, Martin (2001). Representations and Characters of Groups (2nd ed.). Cambridge University Press. ISBN 0-521-00392-X. • Serre, Jean-Pierre (1977). Linear Representations of Finite Groups. Springer-Verlag. ISBN 0-387-90190-6.

External links • Character (http://planetmath.org/encyclopedia/Character.html) at PlanetMath.

Sylow theorem In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician L. Sylow (1872) that give detailed information about the number of subgroups of fixed order that a given finite group contains. The Sylow theorems form a fundamental part of finite group theory and have very important applications in the classification of finite simple groups. For a prime number p, a Sylow p-subgroup (sometimes p-Sylow subgroup) of a group G is a maximal p-subgroup of G, i.e., a subgroup of G which is a p-group (so that the order of any group element is a power of p), and which is not a proper subgroup of any other p-subgroup of G. The set of all Sylow p-subgroups for a given prime p is sometimes written Sylp(G). The Sylow theorems assert a partial converse to Lagrange's theorem that for any finite group G the order (number of elements) of every subgroup of G divides the order of G. For any prime factor p of the order of a finite group G, there exists a Sylow p-subgroup of G. The order of a Sylow p-subgroup of a finite group G is pn, where n is the multiplicity of p in the order of G, and any subgroup of order pn is a Sylow p-subgroup of G. The Sylow p-subgroups of a group (for fixed prime p) are conjugate to each other. The number of Sylow p-subgroups of a group for fixed prime p is congruent to 1 mod p.

Sylow theorems Collections of subgroups which are each maximal in one sense or another are common in group theory. The surprising result here is that in the case of Sylp(G), all members are actually isomorphic to each other and have the largest possible order: if |G| = pnm with where p does not divide m, then any Sylow p-subgroup P has order |P| = pn. That is, P is a p-group and gcd(|G:P|, p) = 1. These properties can be exploited to further analyze the structure of G. The following theorems were first proposed and proven by Ludwig Sylow in 1872, and published in Mathematische Annalen. Theorem 1: For any prime factor p with multiplicity n of the order of a finite group G, there exists a Sylow p-subgroup of G, of order pn. The following weaker version of theorem 1 was first proved by Cauchy.

133

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134

Corollary: Given a finite group G and a prime number p dividing the order of G, then there exists an element of order p in G . Theorem 2: Given a finite group G and a prime number p, all Sylow p-subgroups of G are conjugate (and therefore isomorphic) to each other, i.e. if H and K are Sylow p-subgroups of G, then there exists an element g in G with g−1Hg = K. Theorem 3: Let p be a prime factor with multiplicity n of the order of a finite group G, so that the order of G can be written as pn · m, where n > 0 and p does not divide m. Let np be the number of Sylow p-subgroups of G. Then the following hold: • np divides m, which is the index of the Sylow p-subgroup in G. • np ≡ 1 mod p. • np = |G : NG(P)|, where P is any Sylow p-subgroup of G and NG denotes the normalizer.

Consequences The Sylow theorems imply that for a prime number p every Sylow p-subgroup is of the same order, pn. Conversely, if a subgroup has order pn, then it is a Sylow p-subgroup, and so is isomorphic to every other Sylow p-subgroup. Due to the maximality condition, if H is any p-subgroup of G, then H is a subgroup of a p-subgroup of order pn A very important consequence of Theorem 2 is that the condition np = 1 is equivalent to saying that the Sylow p-subgroup of G is a normal subgroup. (There are groups which have normal subgroups but no normal Sylow subgroups, such as S4.)

Sylow theorems for infinite groups There is an analogue of the Sylow theorems for infinite groups. We define a Sylow p-subgroup in an infinite group to be a p-subgroup (that is, every element in it has p-power order) which is maximal for inclusion among all p-subgroups in the group. Such subgroups exist by Zorn's lemma. Theorem: If K is a Sylow p-subgroup of G, and np = |Cl(K)| is finite, then every Sylow p-subgroup is conjugate to K, and np ≡ 1 mod p, where Cl(K) denotes the conjugacy class of K.

Examples A simple illustration of Sylow subgroups and the Sylow theorems are the dihedral group of the n-gon, For n odd, is the higher power of 2 dividing the order, and thus subgroups of order 2 are Sylow subgroups. These are the groups generated by a reflection, of which there are n, and they are all conjugate under rotations; geometrically the axes of symmetry pass through a vertex and a side. By contrast, if n is even, then 4 divides the order of the group, and these are no longer Sylow subgroups, and in fact they fall into two conjugacy classes, geometrically according to whether they pass through two vertices or two faces. These are related by an outer automorphism, which can be represented by rotation through half the minimal rotation in the dihedral group.

Example applications

In

all reflections are conjugate, as

reflections correspond to Sylow 2-subgroups.

Sylow theorem

In

135

reflections no longer correspond to

Sylow 2-subgroups, and fall into two conjugacy classes.

Cyclic group orders Some numbers n are such that every group of order n is cyclic. One can show that n = 15 is such a number using the Sylow theorems: Let G be a group of order 15 = 3 · 5 and n3 be the number of Sylow 3-subgroups. Then and . The only value satisfying these constraints is 1; therefore, there is only one subgroup of order 3, and it must be normal (since it has no distinct conjugates). Similarly, n5 must divide 3, and n5 must equal 1 (mod 5); thus it must also have a single normal subgroup of order 5. Since 3 and 5 are coprime, the intersection of these two subgroups is trivial, and so G must be the internal direct product of groups of order 3 and 5, that is the cyclic group of order 15. Thus, there is only one group of order 15 (up to isomorphism).

Small groups are not simple A more complex example involves the order of the smallest simple group which is not cyclic. Burnside's paqb theorem states that if the order of a group is the product of two prime powers, then it is solvable, and so the group is not simple, or is of prime order and is cyclic. This rules out every group up to order 30 (= 2 · 3 · 5). If G is simple, and |G| = 30, then n3 must divide 10 ( = 2 · 5), and n3 must equal 1 (mod 3). Therefore n3 = 10, since neither 4 nor 7 divides 10, and if n3 = 1 then, as above, G would have a normal subgroup of order 3, and could not be simple. G then has 10 distinct cyclic subgroups of order 3, each of which has 2 elements of order 3 (plus the identity). This means G has at least 20 distinct elements of order 3. As well, n5 = 6, since n5 must divide 6 ( = 2 · 3), and n5 must equal 1 (mod 5). So G also has 24 distinct elements of order 5. But the order of G is only 30, so a simple group of order 30 cannot exist. Next, suppose |G| = 42 = 2 · 3 · 7. Here n7 must divide 6 ( = 2 · 3) and n7 must equal 1 (mod 7), so n7 = 1. So, as before, G can not be simple. On the other hand for |G| = 60 = 22 · 3 · 5, then n3 = 10 and n5 = 6 is perfectly possible. And in fact, the smallest simple non-cyclic group is A5, the alternating group over 5 elements. It has order 60, and has 24 cyclic permutations of order 5, and 20 of order 3.

Sylow theorem

Fusion results Frattini's argument shows that a Sylow subgroup of a normal subgroup provides a factorization of a finite group. A slight generalization known as Burnside's fusion theorem states that if G is a finite group with Sylow p-subgroup P and two subsets A and B normalized by P, then A and B are G-conjugate if and only if they are NG(P)-conjugate. The proof is a simple application of Sylow's theorem: If B=Ag, then the normalizer of B contains not only P but also Pg (since Pg is contained in the normalizer of Ag). By Sylow's theorem P and Pg are conjugate not only in G, but in the normalizer of B. Hence gh−1 normalizes P for some h that normalizes B, and then Agh−1 = Bh−1 = B, so that A and B are NG(P)-conjugate. Burnside's fusion theorem can be used to give a more power factorization called a semidirect product: if G is a finite group whose Sylow p-subgroup P is contained in the center of its normalizer, then G has a normal subgroup K of order coprime to P, G = PK and P∩K = 1, that is, G is p-nilpotent. Less trivial applications of the Sylow theorems include the focal subgroup theorem, which studies the control a Sylow p-subgroup of the derived subgroup has on the structure of the entire group. This control is exploited at several stages of the classification of finite simple groups, and for instance defines the case divisions used in the Alperin–Brauer–Gorenstein theorem classifying finite simple groups whose Sylow 2-subgroup is a quasi-dihedral group. These rely on J. L. Alperin's strengthening of the conjugacy portion of Sylow's theorem to control what sorts of elements are used in the conjugation.

Proof of the Sylow theorems The Sylow theorems have been proved in a number of ways, and the history of the proofs themselves are the subject of many papers including (Waterhouse 1979), (Scharlau 1988), (Casadio & Zappa 1990), (Gow 1994), and to some extent (Meo 2004). One proof of the Sylow theorems exploit the notion of group action in various creative ways. The group G acts on itself or on the set of its p-subgroups in various ways, and each such action can be exploited to prove one of the Sylow theorems. The following proofs are based on combinatorial arguments of (Wielandt 1959). In the following, we use a | b as notation for "a divides b" and a b for the negation of this statement. Theorem 1: A finite group G whose order |G| is divisible by a prime power pk has a subgroup of order pk. Proof: Let |G| = pkm = pk+ru such that p does not divide u, and let Ω denote the set of subsets of G of size pk. G acts on Ω by left multiplication. The orbits Gω = {gω | g ∈ G} of the ω ∈ Ω are the equivalence classes under the action of G. For any ω ∈ Ω consider its stabilizer subgroup Gω. For any fixed element α ∈ ω the function [g ↦ gα] maps Gω to ω injectively: for any two g,h ∈ Gω we have that gα = hα implies g = h, because α ∈ ω ⊆ G means that one may cancel on the right. Therefore pk = |ω| ≥ |Gω|. On the other hand

and no power of p remains in any of the factors inside the product on the right. Hence νp(|Ω|) = νp(m) = r. Let R ⊆ Ω be a complete representation of all the equivalence classes under the action of G. Then,

Thus, there exists an element ω ∈ R such that s := νp(|Gω|) ≤ νp(|Ω|) = r. Hence |Gω| = psv where p does not divide v. By the stabilizer-orbit-theorem we have |Gω| = |G| / |Gω| = pk+r-su / v. Therefore pk | |Gω|, so pk ≤ |Gω| and Gω is the desired subgroup.

136

Sylow theorem Lemma: Let G be a finite p-group, let G act on a finite set Ω, and let Ω0 denote the set of points of Ω that are fixed under the action of G. Then |Ω| ≡ |Ω0| mod p. Proof: Write Ω as a disjoint sum of its orbits under G. Any element x ∈ Ω not fixed by G will lie in an orbit of order |G|/|Gx| (where Gx denotes the stabilizer), which is a multiple of p by assumption. The result follows immediately. Theorem 2: If H is a p-subgroup of G and P is a Sylow p-subgroup of G, then there exists an element g in G such that g−1Hg ≤ P. In particular, all Sylow p-subgroups of G are conjugate to each other (and therefore isomorphic), i.e. if H and K are Sylow p-subgroups of G, then there exists an element g in G with g−1Hg = K. Proof: Let Ω be the set of left cosets of P in G and let H act on Ω by left multiplication. Applying the Lemma to H on Ω, we see that |Ω0| ≡ |Ω| = [G : P] mod p. Now p [G : P] by definition so p |Ω0|, hence in particular |Ω0| ≠ 0 so there exists some gP ∈ Ω0. It follows that for some g ∈ G and ∀ h ∈ H we have hgP = gP so g−1hgP ⊆ P and therefore g−1Hg ≤ P. Now if H is a Sylow p-subgroup, |H| = |P| = |gPg−1| so that H = gPg−1 for some g ∈ G.

Theorem 3: Let q denote the order of any Sylow p-subgroup of a finite group G. Then np | |G|/q and np ≡ 1 mod p. Proof: By Theorem 2, np = [G : NG(P)], where P is any such subgroup, and NG(P) denotes the normalizer of P in G, so this number is a divisor of |G|/q. Let Ω be the set of all Sylow p-subgroups of G, and let P act on Ω by conjugation. Let Q ∈ Ω0 and observe that then Q = xQx−1 for all x ∈ P so that P ≤ NG(Q). By Theorem 2, P and Q are conjugate in NG(Q) in particular, and Q is normal in NG(Q), so then P = Q. It follows that Ω0 = {P} so that, by the Lemma, |Ω| ≡ |Ω0| = 1 mod p.

Algorithms The problem of finding a Sylow subgroup of a given group is an important problem in computational group theory. One proof of the existence of Sylow p-subgroups is constructive: if H is a p-subgroup of G and the index [G:H] is divisible by p, then the normalizer N = NG(H) of H in G is also such that [N:H] is divisible by p. In other words, a polycyclic generating system of a Sylow p-subgroup can be found by starting from any p-subgroup H (including the identity) and taking elements of p-power order contained in the normalizer of H but not in H itself. The algorithmic version of this (and many improvements) is described in textbook form in (Butler 1991, Chapter 16), including the algorithm described in (Cannon 1971). These versions are still used in the GAP computer algebra system. In permutation groups, it has been proven in (Kantor 1985a, 1985b, 1988, 1990) that a Sylow p-subgroup and its normalizer can be found in polynomial time of the input (the degree of the group times the number of generators). These algorithms are described in textbook form in (Seress 2003), and are now becoming practical as the constructive recognition of finite simple groups becomes a reality. In particular, versions of this algorithm are used in the Magma computer algebra system.

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See also • Frattini's argument • Hall subgroup • Maximal subgroup

Notes References • Sylow, L. (1872), "Théorèmes sur les groupes de substitutions", Mathematische Annalen 5: 584–594, doi:10.1007/BF01442913

Proofs • Casadio, Giuseppina; Zappa, Guido (1990), "History of the Sylow theorem and its proofs", Bollettino di Storia delle Scienze Matematiche 10 (1): 29–75, MR1096350, ISSN 0392-4432 • Gow, Rod (1994), "Sylow's proof of Sylow's theorem", Irish Mathematical Society Bulletin (33): 55–63, MR1313412, ISSN 0791-5578 • Kammüller, Florian; Paulson, Lawrence C. (1999), "A formal proof of Sylow's theorem. An experiment in abstract algebra with Isabelle HOL" (http://www.cl.cam.ac.uk/users/lcp/papers/Kammueller/sylow.pdf), Journal of Automated Reasoning 23 (3): 235–264, doi:10.1023/A:1006269330992, MR1721912, ISSN 0168-7433 • Meo, M. (2004), "The mathematical life of Cauchy's group theorem", Historia Mathematica 31 (2): 196–221, doi:10.1016/S0315-0860(03)00003-X, MR2055642, ISSN 0315-0860 • Scharlau, Winfried (1988), "Die Entdeckung der Sylow-Sätze", Historia Mathematica 15 (1): 40–52, doi:10.1016/0315-0860(88)90048-1, MR931678, ISSN 0315-0860 • Waterhouse, William C. (1979), "The early proofs of Sylow's theorem", Archive for History of Exact Sciences 21 (3): 279–290, doi:10.1007/BF00327877, MR575718, ISSN 0003-9519 • Wielandt, Helmut (1959), "Ein Beweis für die Existenz der Sylowgruppen", Archiv der Mathematik 10: 401–402, doi:10.1007/BF01240818, MR0147529, ISSN 0003-9268

Algorithms • Butler, G. (1991), Fundamental algorithms for permutation groups, Lecture Notes in Computer Science, 559, Berlin, New York: Springer-Verlag, MR1225579, ISBN 978-3-540-54955-0 • Cannon, John J. (1971), "Computing local structure of large finite groups", Computers in algebra and number theory (Proc. SIAM-AMS Sympos. Appl. Math., New York, 1970), Providence, R.I.: Amer. Math. Soc., pp. 161–176, MR0367027 • Kantor, William M. (1985), "Polynomial-time algorithms for finding elements of prime order and Sylow subgroups", Journal of Algorithms 6 (4): 478–514, MR813589, ISSN 0196-6774 • Kantor, William M. (1985), "Sylow's theorem in polynomial time", Journal of Computer and System Sciences 30 (3): 359–394, doi:10.1016/0022-0000(85)90052-2, MR805654, ISSN 1090-2724 • Kantor, William M.; Taylor, Donald E. (1988), "Polynomial-time versions of Sylow's theorem", Journal of Algorithms 9 (1): 1–17, MR925595, ISSN 0196-6774 • Kantor, William M. (1990), "Finding Sylow normalizers in polynomial time", Journal of Algorithms 11 (4): 523–563, MR1079450, ISSN 0196-6774 • Seress, Ákos (2003), Permutation group algorithms, Cambridge Tracts in Mathematics, 152, Cambridge University Press, MR1970241, ISBN 978-0-521-66103-4

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Lie algebra In mathematics, a Lie algebra (pronounced /ˈliː/ ("lee"), not /ˈlaɪ/ ("lye")) is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" (after Sophus Lie) was introduced by Hermann Weyl in the 1930s. In older texts, the name "infinitesimal group" is used.

Definition and first properties A Lie algebra is a vector space

over some field F together with a binary operation [·, ·]

called the Lie bracket, which satisfies the following axioms: • Bilinearity:

for all scalars a, b in F and all elements x, y, z in • Alternating on

.

:

for all x in . This implies anticommutativity, or skew-symmetry (in fact the conditions are equivalent for any Lie algebra over any field whose characteristic is not 2):

for all elements x, y in

.

• The Jacobi identity:

for all x, y, z in

.

For any associative algebra A with multiplication , one can construct a Lie algebra L(A). As a vector space, L(A) is the same as A. The Lie bracket of two elements of L(A) is defined to be their commutator in A:

The associativity of the multiplication * in A implies the Jacobi identity of the commutator in L(A). In particular, the associative algebra of n × n matrices over a field F gives rise to the general linear Lie algebra The associative algebra A is called an enveloping algebra of the Lie algebra L(A). It is known that every Lie algebra can be embedded into one that arises from an associative algebra in this fashion. See universal enveloping algebra.

Homomorphisms, subalgebras, and ideals The Lie bracket is not an associative operation in general, meaning that

need not equal

.

Nonetheless, much of the terminology that was developed in the theory of associative rings or associative algebras is commonly applied to Lie algebras. A subspace subalgebra. If a subspace

that is closed under the Lie bracket is called a Lie

satisfies a stronger condition that

then I is called an ideal in the Lie algebra .[1] A Lie algebra in which the commutator is not identically zero and which has no proper ideals is called simple. A homomorphism between two Lie algebras (over the same ground field) is a linear map that is compatible with the commutators:

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140

for all elements x and y in . As in the theory of associative rings, ideals are precisely the kernels of homomorphisms, given a Lie algebra and an ideal I in it, one constructs the factor algebra , and the first isomorphism theorem holds for Lie algebras. Given two Lie algebras consisting of the pairs

and

, their direct sum is the vector space

, with the operation

Examples • Any vector space V endowed with the identically zero Lie bracket becomes a Lie algebra. Such Lie algebras are called abelian, cf. below. Any one-dimensional Lie algebra over a field is abelian, by the antisymmetry of the Lie bracket. • The three-dimensional Euclidean space R3 with the Lie bracket given by the cross product of vectors becomes a three-dimensional Lie algebra. • The Heisenberg algebra is a three-dimensional Lie algebra with generators (see also the definition at Generating set):

whose commutation relations are

It is explicitly exhibited as the space of 3×3 strictly upper-triangular matrices. • The subspace of the general linear Lie algebra

consisting of matrices of trace zero is a subalgebra,[2] the

special linear Lie algebra, denoted • Any Lie group G defines an associated real Lie algebra

. The definition in general is somewhat

technical, but in the case of real matrix groups, it can be formulated via the exponential map, or the matrix exponent. The Lie algebra

consists of those matrices X for which

for all real numbers t. The Lie bracket of is given by the commutator of matrices. As a concrete example, consider the special linear group SL(n,R), consisting of all n × n matrices with real entries and determinant 1. This is a matrix Lie group, and its Lie algebra consists of all n × n matrices with real entries and trace 0. • The real vector space of all n × n skew-hermitian matrices is closed under the commutator and forms a real Lie algebra denoted . This is the Lie algebra of the unitary group U(n). • An important class of infinite-dimensional real Lie algebras arises in differential topology. The space of smooth vector fields on a differentiable manifold M forms a Lie algebra, where the Lie bracket is defined to be the commutator of vector fields. One way of expressing the Lie bracket is through the formalism of Lie derivatives, which identifies a vector field X with a first order partial differential operator LX acting on smooth functions by letting LX(f) be the directional derivative of the function f in the direction of X. The Lie bracket [X,Y] of two vector fields is the vector field defined through its action on functions by the formula:

This Lie algebra is related to the pseudogroup of diffeomorphisms of M. • The commutation relations between the x, y, and z components of the angular momentum operator in quantum mechanics form a representation of a complex three-dimensional Lie algebra, which is the complexification of the Lie algebra so(3) of the three-dimensional rotation group:

Lie algebra

141

• Kac–Moody algebra is an example of an infinite-dimensional Lie algebra.

Structure theory and classification Every finite-dimensional real or complex Lie algebra has a faithful representation by matrices (Ado's theorem). Lie's fundamental theorems describe a relation between Lie groups and Lie algebras. In particular, any Lie group gives rise to a canonically determined Lie algebra (concretely, the tangent space at the identity), and conversely, for any Lie algebra there is a corresponding connected Lie group (Lie's third theorem). This Lie group is not determined uniquely, however, any two connected Lie groups with the same Lie algebra are locally isomorphic, and in particular, have the same universal cover. For instance, the special orthogonal group SO(3) and the special unitary group SU(2) give rise to the same Lie algebra, which is isomorphic to R3 with the cross-product, and SU(2) is a simply-connected twofold cover of SO(3). Real and complex Lie algebras can be classified to some extent, and this is often an important step toward the classification of Lie groups.

Abelian, nilpotent, and solvable Analogously to abelian, nilpotent, and solvable groups, defined in terms of the derived subgroups, one can define abelian, nilpotent, and solvable Lie algebras. A Lie algebra is abelian if the Lie bracket vanishes, i.e. [x,y] = 0, for all x and y in . Abelian Lie algebras correspond to commutative (or abelian) connected Lie groups such as vector spaces or tori and are all of the form

meaning an n-dimensional vector space with the trivial Lie bracket.

A more general class of Lie algebras is defined by the vanishing of all commutators of given length. A Lie algebra is nilpotent if the lower central series

becomes zero eventually. By Engel's theorem, a Lie algebra is nilpotent if and only if for every u in endomorphism

the adjoint

is nilpotent. More generally still, a Lie algebra

is said to be solvable if the derived series:

becomes zero eventually. Every finite-dimensional Lie algebra has a unique maximal solvable ideal, called its radical. Under the Lie correspondence, nilpotent (respectively, solvable) connected Lie groups correspond to nilpotent (respectively, solvable) Lie algebras.

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Simple and semisimple A Lie algebra is "simple" if it has no non-trivial ideals and is not abelian. A Lie algebra is called semisimple if its radical is zero. Equivalently, is semisimple if it does not contain any non-zero abelian ideals. In particular, a simple Lie algebra is semisimple. Conversely, it can be proven that any semisimple Lie algebra is the direct sum of its minimal ideals, which are canonically determined simple Lie algebras. The concept of semisimplicity for Lie algebras is closely related with the complete reducibility of their representations. When the ground field F has characteristic zero, semisimplicity of a Lie algebra over F is equivalent to the complete reducibility of all finite-dimensional representations of An early proof of this statement proceeded via connection with compact groups (Weyl's unitary trick), but later entirely algebraic proofs were found.

Classification In many ways, the classes of semisimple and solvable Lie algebras are at the opposite ends of the full spectrum of the Lie algebras. The Levi decomposition expresses an arbitrary Lie algebra as a semidirect sum of its solvable radical and a semisimple Lie algebra, almost in a canonical way. Semisimple Lie algebras over an algebraically closed field have been completely classified through their root systems. The classification of solvable Lie algebras is a 'wild' problem, and cannot be accomplished in general. Cartan's criterion gives conditions for a Lie algebra to be nilpotent, solvable, or semisimple. It is based on the notion of the Killing form, a symmetric bilinear form on defined by the formula

where tr denotes the trace of a linear operator. A Lie algebra nondegenerate. A Lie algebra is solvable if and only if

is semisimple if and only if the Killing form is

Relation to Lie groups Although Lie algebras are often studied in their own right, historically they arose as a means to study Lie groups. Given a Lie group, a Lie algebra can be associated to it either by endowing the tangent space to the identity with the differential of the adjoint map, or by considering the left-invariant vector fields as mentioned in the examples. This association is functorial, meaning that homomorphisms of Lie groups lift to homomorphisms of Lie algebras, and various properties are satisfied by this lifting: it commutes with composition, it maps Lie subgroups, kernels, quotients and cokernels of Lie groups to subalgebras, kernels, quotients and cokernels of Lie algebras, respectively. The functor which takes each Lie group to its Lie algebra and each homomorphism to its differential is a faithful and exact functor. This functor is not invertible; different Lie groups may have the same Lie algebra, for example SO(3) and SU(2) have isomorphic Lie algebras. Even worse, some Lie algebras need not have any associated Lie group. Nevertheless, when the Lie algebra is finite-dimensional, there is always at least one Lie group whose Lie algebra is the one under discussion, and a preferred Lie group can be chosen. Any finite-dimensional connected Lie group has a universal cover. This group can be constructed as the image of the Lie algebra under the exponential map. More generally, we have that the Lie algebra is homeomorphic to a neighborhood of the identity. But globally, if the Lie group is compact, the exponential will not be injective, and if the Lie group is not connected, simply connected or compact, the exponential map need not be surjective. If the Lie algebra is infinite-dimensional, the issue is more subtle. In many instances, the exponential map is not even locally a homeomorphism (for example, in Diff(S1), one may find diffeomorphisms arbitrarily close to the identity which are not in the image of exp). Furthermore, some infinite-dimensional Lie algebras are not the Lie algebra of any group. The correspondence between Lie algebras and Lie groups is used in several ways, including in the classification of Lie groups and the related matter of the representation theory of Lie groups. Every representation of a Lie algebra

Lie algebra lifts uniquely to a representation of the corresponding connected, simply connected Lie group, and conversely every representation of any Lie group induces a representation of the group's Lie algebra; the representations are in one to one correspondence. Therefore, knowing the representations of a Lie algebra settles the question of representations of the group. As for classification, it can be shown that any connected Lie group with a given Lie algebra is isomorphic to the universal cover mod a discrete central subgroup. So classifying Lie groups becomes simply a matter of counting the discrete subgroups of the center, once the classification of Lie algebras is known (solved by Cartan et al. in the semisimple case).

Category theoretic definition Using the language of category theory, a Lie algebra can be defined as an object A in Vec, the category of vector spaces together with a morphism [.,.]: A ⊗ A → A, where ⊗ refers to the monoidal product of Vec, such that • • where τ (a ⊗ b) := b ⊗ a and σ is the cyclic permutation braiding (id ⊗ τA,A) ° (τA,A ⊗ id). In diagrammatic form:

Notes [1] Due to the anticommutativity of the commutator, the notions of a left and right ideal in a Lie algebra coincide. [2] Humphreys p.2

References • Hall, Brian C. Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Springer, 2003. ISBN 0-387-40122-9 • Erdmann, Karin & Wildon, Mark. Introduction to Lie Algebras, 1st edition, Springer, 2006. ISBN 1-84628-040-0 • Humphreys, James E. Introduction to Lie Algebras and Representation Theory, Second printing, revised. Graduate Texts in Mathematics, 9. Springer-Verlag, New York, 1978. ISBN 0-387-90053-5 • Jacobson, Nathan, Lie algebras, Republication of the 1962 original. Dover Publications, Inc., New York, 1979. ISBN 0-486-63832-4 • Kac, Victor G. et al. Course notes for MIT 18.745: Introduction to Lie Algebras, http://www-math.mit.edu/ ~lesha/745lec/ • O'Connor, J. J. & Robertson, E.F. Biography of Sophus Lie, MacTutor History of Mathematics Archive, http:// www-history.mcs.st-and.ac.uk/Biographies/Lie.html • O'Connor, J. J. & Robertson, E.F. Biography of Wilhelm Killing, MacTutor History of Mathematics Archive, http://www-history.mcs.st-and.ac.uk/Biographies/Killing.html • Steeb, W.-H. Continuous Symmetries, Lie Algebras, Differential Equations and Computer Algebra, second edition, World Scientific, 2007, ISBN 978-981-270-809-0 • Varadarajan, V. S. Lie Groups, Lie Algebras, and Their Representations, 1st edition, Springer, 2004. ISBN 0-387-90969-9

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Class group In mathematics, the extent to which unique factorization fails in the ring of integers of an algebraic number field (or more generally any Dedekind domain) can be described by a certain group known as an ideal class group (or class group). If this group is finite (as it is in the case of the ring of integers of a number field), then the order of the group is called the class number. The multiplicative theory of a Dedekind domain is intimately tied to the structure of its class group. For example, the class group of a Dedekind domain is trivial if and only if the ring is a unique factorization domain.

History and origin of the ideal class group Ideal class groups (or, rather, what were effectively ideal class groups) were studied some time before the idea of an ideal was formulated. These groups appeared in the theory of quadratic forms: in the case of binary integral quadratic forms, as put into something like a final form by Gauss, a composition law was defined on certain equivalence classes of forms. This gave a finite abelian group, as was recognised at the time. Later Kummer was working towards a theory of cyclotomic fields. It had been realised (probably by several people) that failure to complete proofs in the general case of Fermat's last theorem by factorisation using the roots of unity was for a very good reason: a failure of the fundamental theorem of arithmetic to hold in the rings generated by those roots of unity was a major obstacle. Out of Kummer's work for the first time came a study of the obstruction to the factorisation. We now recognise this as part of the ideal class group: in fact Kummer had isolated the p-torsion in that group for the field of p-roots of unity, for any prime number p, as the reason for the failure of the standard method of attack on the Fermat problem (see regular prime). Somewhat later again Dedekind formulated the concept of ideal, Kummer having worked in a different way. At this point the existing examples could be unified. It was shown that while rings of algebraic integers do not always have unique factorization into primes (because they need not be principal ideal domains), they do have the property that every proper ideal admits a unique factorization as a product of prime ideals (that is, every ring of algebraic integers is a Dedekind domain). The size of the ideal class group can be considered as a measure for the deviation of a ring from being a principal domain; a ring is a principal domain if and only if it has a trivial ideal class group.

Technical development If R is an integral domain, define a relation ~ on nonzero fractional ideals of R by I ~ J whenever there exist nonzero elements a and b of R such that (a)I = (b)J. (Here the notation (a) means the principal ideal of R consisting of all the multiples of a.) It is easily shown that this is an equivalence relation. The equivalence classes are called the ideal classes of R. Ideal classes can be multiplied: if [I] denotes the equivalence class of the ideal I, then the multiplication [I][J] = [IJ] is well-defined and commutative. The principal ideals form the ideal class [R] which serves as an identity element for this multiplication. Thus a class [I] has an inverse [J] if and only if there is an ideal J such that IJ is a principal ideal. In general, such a J may not exist and consequently the set of ideal classes of R may only be a monoid. However, if R is the ring of algebraic integers in an algebraic number field, or more generally a Dedekind domain, the multiplication defined above turns the set of fractional ideal classes into an abelian group, the ideal class group of R. The group property of existence of inverse elements follows easily from the fact that, in a Dedekind domain, every non-zero ideal (except R) is a product of prime ideals.

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Properties The ideal class group is trivial (i.e. has only one element) if and only if all ideals of R are principal. In this sense, the ideal class group measures how far R is from being a principal ideal domain, and hence from satisfying unique prime factorization (Dedekind domains are unique factorization domains if and only if they are principal ideal domains). The number of ideal classes (the class number of R) may be infinite in general. In fact, every abelian group is isomorphic to the ideal class group of some Dedekind domain.[1] But if R is in fact a ring of algebraic integers, then the class number is always finite. This is one of the main results of classical algebraic number theory. Computation of the class group is hard, in general; it can be done by hand for the ring of integers in an algebraic number field of small discriminant, using Minkowski's bound. This result gives a bound, depending on the ring, such that every ideal class contains an ideal of norm less than the bound. In general the bound is not sharp enough to make the calculation practical for fields with large discriminant, but computers are well suited to the task. The mapping from rings of integers R to their corresponding class groups is functorial, and the class group can be subsumed under the heading of algebraic K-theory, with K0(R) being the functor assigning to R its ideal class group; more precisely, K0(R) = Z×C(R), where C(R) is the class group. Higher K groups can also be employed and interpreted arithmetically in connection to rings of integers.

Relation with the group of units It was remarked above that the ideal class group provides part of the answer to the question of how much ideals in a Dedekind domain behave like elements. The other part of the answer is provided by the multiplicative group of units of the Dedekind domain, since passage from principal ideals to their generators requires the use of units (and this is the rest of the reason for introducing the concept of fractional ideal, as well): Define a map from K× to the set of all nonzero fractional ideals of R by sending every element to the principal (fractional) ideal it generates. This is a group homomorphism; its kernel is the group of units of R, and its cokernel is the ideal class group of R. The failure of these groups to be trivial is a measure of the failure of the map to be an isomorphism: that is the failure of ideals to act like ring elements, that is to say, like numbers.

Examples of ideal class groups • The rings Z, Z[ω], and Z[i], where ω is a cube root of 1 and i is a fourth root of 1 (i.e. a square root of −1), are all principal ideal domains, and so have class number 1: that is, they have trivial ideal class groups. • If k is a field, then the polynomial ring k[X1, X2, X3, ...] is an integral domain. It has a countably infinite set of ideal classes.

Class numbers of quadratic fields If d is a square-free integer (a product of distinct primes) other than 1, then Q(√d) is a quadratic extension of Q. If d < 0, then the class number of the ring R of algebraic integers of Q(√d) is equal to 1 for precisely the following values of d: d = −1, −2, −3, −7, −11, −19, −43, −67, and −163. This result was first conjectured by Gauss and proven by Kurt Heegner, although Heegner's proof was not believed until Harold Stark gave a later proof in 1967. (See Stark-Heegner theorem.) This is a special case of the famous class number problem. If, on the other hand, d > 0, then it is unknown whether there are infinitely many fields Q(√d) with class number 1. Computational results indicate that there are a great many such fields. However, it is not even known if there are infinitely many number fields with class number 1.[2] For d < 0, the ideal class group of Q(√d) is isomorphic to the class group of integral binary quadratic forms of discriminant equal to the discriminant of Q(√d). For d > 0, the ideal class group may be half the size since the class group of integral binary quadratic forms is isomorphic to the narrow class group of Q(√d).[3]

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Class group Example of a non-trivial class group The quadratic integer ring R = Z [√−5] is the ring of integers of Q(√−5). It does not possess unique factorization; in fact the class group of R is cyclic of order 2. Indeed, the ideal J = (2, 1 + √−5) is not principal, which can be proved by contradiction as follows. If J were generated by an element x of R, then x would divide both 2 and 1 + √−5. Then the norm N(x) of x would divide both N(2) = 4 and N(1 + √−5) = 6, so N(x) would divide 2. We are assuming that x is not a unit of R, so N(x) cannot be 1. It cannot be 2 either, because R has no elements of norm 2, that is, the equation b2 + 5c2 = 2 has no solutions in integers. One also computes that J2 = (2), which is principal, so the class of J in the ideal class group has order two. Showing that there aren't any other ideal classes requires more effort. The fact that this J is not principal is also related to the fact that the element 6 has two distinct factorisations into irreducibles: 6 = 2 × 3 = (1 + √−5) × (1 − √−5).

Connections to class field theory Class field theory is a branch of algebraic number theory which seeks to classify all the abelian extensions of a given algebraic number field, meaning Galois extensions with abelian Galois group. A particularly beautiful example is found in the Hilbert class field of a number field, which can be defined as the maximal unramified abelian extension of such a field. The Hilbert class field L of a number field K is unique and has the following properties: • Every ideal of the ring of integers of K becomes principal in L, i.e., if I is an integral ideal of K then the image of I is a principal ideal in L. • L is a Galois extension of K with Galois group isomorphic to the ideal class group of K. Neither property is particularly easy to prove.

See also • • • • • • • • •

Class number formula Class number problem List of number fields with class number one Principal ideal domain Algebraic K-theory Galois theory Fermat's last theorem Narrow class group Picard group—a generalisation of the class group appearing in algebraic geometry

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Notes [1] Claborn 1966 [2] Neukirch 1999 [3] Fröhlich & Taylor 1993, Theorem 58

References • Claborn, Luther (1966), "Every abelian group is a class group" (http://projecteuclid.org/DPubS?verb=Display& version=1.0&service=UI&handle=euclid.pjm/1102994263&page=record), Pacific Journal of Mathematics 18: 219–222 • Fröhlich, Albrecht; Taylor, Martin (1993), Algebraic number theory, Cambridge Studies in Advanced Mathematics, 27, Cambridge University Press, MR1215934, ISBN 978-0-521-43834-6 • Neukirch, Jürgen (1999), Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften, 322, Berlin: Springer-Verlag, MR1697859, ISBN 978-3-540-65399-8

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Abelian group

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Abelian group Concepts in group theory category of groups subgroups, normal subgroups group homomorphisms, kernel, image, quotient direct product, direct sum semidirect product, wreath product Types of groups simple, finite, infinite discrete, continuous multiplicative, additive cyclic, abelian, dihedral nilpotent, solvable list of group theory topics glossary of group theory

An abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order (the axiom of commutativity). Abelian groups generalize the arithmetic of addition of integers. They are named after Niels Henrik Abel.[1] The concept of an abelian group is one of the first concepts encountered in undergraduate abstract algebra, with many other basic objects, such as a module and a vector space, being its refinements. The theory of abelian groups is generally simpler than that of their non-abelian counterparts, and finite abelian groups are very well understood. On the other hand, the theory of infinite abelian groups is an area of current research.

Definition An abelian group is a set, A, together with an operation "•" that combines any two elements a and b to form another element denoted a • b. The symbol "•" is a general placeholder for a concretely given operation. To qualify as an abelian group, the set and operation, (A, •), must satisfy five requirements known as the abelian group axioms: Closure For all a, b in A, the result of the operation a • b is also in A. Associativity For all a, b and c in A, the equation (a • b) • c = a • (b • c) holds. Identity element There exists an element e in A, such that for all elements a in A, the equation e • a = a • e = a holds. Inverse element For each a in A, there exists an element b in A such that a • b = b • a = e, where e is the identity element. Commutativity For all a, b in A, a • b = b • a. More compactly, an abelian group is a commutative group. A group in which the group operation is not commutative is called a "non-abelian group" or "non-commutative group".

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Facts Notation There are two main notational conventions for abelian groups — additive and multiplicative. Convention Addition Multiplication

Operation

Identity

Powers

Inverse

x+y

0

nx

−x

x * y or xy

e or 1

xn

x −1

Generally, the multiplicative notation is the usual notation for groups, while the additive notation is the usual notation for modules. The additive notation may also be used to emphasize that a particular group is abelian, whenever both abelian and non-abelian groups are considered.

Multiplication table To verify that a finite group is abelian, a table (matrix) - known as a Cayley table - can be constructed in a similar fashion to a multiplication table. If the group is G = {g1 = e, g2, ..., gn} under the operation ⋅, the (i, j)'th entry of this table contains the product gi ⋅ gj. The group is abelian if and only if this table is symmetric about the main diagonal (i.e. if the matrix is a symmetric matrix). This is true since if the group is abelian, then gi ⋅ gj = gj ⋅ gi. This implies that the (i, j)'th entry of the table equals the (j, i)'th entry - i.e. the table is symmetric about the main diagonal.

Examples • For the integers and the operation addition "+", denoted (Z,+), the operation + combines any two integers to form a third integer, addition is associative, zero is the additive identity, every integer n has an additive inverse, −n, and the addition operation is commutative since m + n = n + m for any two integers m and n. • Every cyclic group G is abelian, because if x, y are in G, then xy = aman = am + n = an + m = anam = yx. Thus the integers, Z, form an abelian group under addition, as do the integers modulo n, Z/nZ. • Every ring is an abelian group with respect to its addition operation. In a commutative ring the invertible elements, or units, form an abelian multiplicative group. In particular, the real numbers are an abelian group under addition, and the nonzero real numbers are an abelian group under multiplication. • Every subgroup of an abelian group is normal, so each subgroup gives rise to a quotient group. Subgroups, quotients, and direct sums of abelian groups are again abelian. In general, matrices, even invertible matrices, do not form an abelian group under multiplication because matrix multiplication is generally not commutative. However, some groups of matrices are abelian groups under matrix multiplication - one example is the group of 2x2 rotation matrices.

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Historical remarks Abelian groups were named for Norwegian mathematician Niels Henrik Abel by Camille Jordan because Abel found that the commutativity of the group of an equation implies its roots are solvable by radicals. See Section 6.5 of Cox (2004) for more information on the historical background.

Properties If n is a natural number and x is an element of an abelian group G written additively, then nx can be defined as x + x + ... + x (n summands) and (−n)x = −(nx). In this way, G becomes a module over the ring Z of integers. In fact, the modules over Z can be identified with the abelian groups. Theorems about abelian groups (i.e. modules over the principal ideal domain Z) can often be generalized to theorems about modules over an arbitrary principal ideal domain. A typical example is the classification of finitely generated abelian groups which is a specialization of the structure theorem for finitely generated modules over a principal ideal domain. In the case of finitely generated abelian groups, this theorem guarantees that an abelian group splits as a direct sum of a torsion group and a free abelian group. The former may be written as a direct sum of finitely many groups of the form Z/pkZ for p prime, and the latter is a direct sum of finitely many copies of Z. If f, g : G  →  H are two group homomorphisms between abelian groups, then their sum f + g, defined by (f + g)(x) = f(x) + g(x), is again a homomorphism. (This is not true if H is a non-abelian group.) The set Hom(G, H) of all group homomorphisms from G to H thus turns into an abelian group in its own right. Somewhat akin to the dimension of vector spaces, every abelian group has a rank. It is defined as the cardinality of the largest set of linearly independent elements of the group. The integers and the rational numbers have rank one, as well as every subgroup of the rationals.

Finite abelian groups Cyclic groups of integers modulo n, Z/nZ, were among the first examples of groups. It turns out that an arbitrary finite abelian group is isomorphic to a direct sum of finite cyclic groups of prime power order, and these orders are uniquely determined, forming a complete system of invariants. The automorphism group of a finite abelian group can be described directly in terms of these invariants. The theory had been first developed in the 1879 paper of Georg Frobenius and Ludwig Stickelberger and later was both simplified and generalized to finitely generated modules over a principal ideal domain, forming an important chapter of linear algebra.

Classification The fundamental theorem of finite abelian groups states that every finite abelian group G can be expressed as the direct sum of cyclic subgroups of prime-power order. This is a special case of the fundamental theorem of finitely generated abelian groups when G has zero rank. The cyclic group

of order mn is isomorphic to the direct sum of

and

if and only if m and n are

coprime. It follows that any finite abelian group G is isomorphic to a direct sum of the form in either of the following canonical ways: • the numbers k1,...,ku are powers of primes • k1 divides k2, which divides k3, and so on up to ku. For example,

can be expressed as the direct sum of two cyclic subgroups of order 3 and 5: . The same can be said for any abelian group of order 15, leading to the

remarkable conclusion that all abelian groups of order 15 are isomorphic.

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For another example, every abelian group of order 8 is isomorphic to either modulo 8),

(the integers 0 to 7 under addition

(the odd integers 1 to 15 under multiplication modulo 16), or

.

See also list of small groups for finite abelian groups of order 16 or less.

Automorphisms One can apply the fundamental theorem to count (and sometimes determine) the automorphisms of a given finite abelian group G. To do this, one uses the fact (which will not be proved here) that if G splits as a direct sum H K of subgroups of coprime order, then Aut(H

K)

Aut(H)

Aut(K).

Given this, the fundamental theorem shows that to compute the automorphism group of G it suffices to compute the automorphism groups of the Sylow p-subgroups separately (that is, all direct sums of cyclic subgroups, each with order a power of p). Fix a prime p and suppose the exponents ei of the cyclic factors of the Sylow p-subgroup are arranged in increasing order:

for some n > 0. One needs to find the automorphisms of

One special case is when n = 1, so that there is only one cyclic prime-power factor in the Sylow p-subgroup P. In this case the theory of automorphisms of a finite cyclic group can be used. Another special case is when n is arbitrary but ei = 1 for 1 ≤ i ≤ n. Here, one is considering P to be of the form so elements of this subgroup can be viewed as comprising a vector space of dimension n over the finite field of p elements . The automorphisms of this subgroup are therefore given by the invertible linear transformations, so

where GL is the appropriate general linear group. This is easily shown to have order

In the most general case, where the ei and n are arbitrary, the automorphism group is more difficult to determine. It is known, however, that if one defines

and

then one has in particular dk ≥ k, ck ≤ k, and

One can check that this yields the orders in the previous examples as special cases (see [Hillar,Rhea]).

Abelian group

Infinite abelian groups Тhe simplest infinite abelian group is the infinite cyclic group Z. Any finitely generated abelian group A is isomorphic to the direct sum of r copies of Z and a finite abelian group, which in turn is decomposable into a direct sum of finitely many cyclic groups of primary orders. Even though the decomposition is not unique, the number r, called the rank of A, and the prime powers giving the orders of finite cyclic summands are uniquely determined. By contrast, classification of general infinitely generated abelian groups is far from complete. Divisible groups, i.e. abelian groups A in which the equation nx = a admits a solution x ∈ A for any natural number n and element a of A, constitute one important class of infinite abelian groups that can be completely characterized. Every divisible group is isomorphic to a direct sum, with summands isomorphic to Q and Prüfer groups Qp/Zp for various prime numbers p, and the cardinality of the set of summands of each type is uniquely determined.[2] Moreover, if a divisible group A is a subgroup of an abelian group G then A admits a direct complement: a subgroup C of G such that G = A ⊕ C. Thus divisible groups are injective modules in the category of abelian groups, and conversely, every injective abelian group is divisible (Baer's criterion). An abelian group without non-zero divisible subgroups is called reduced. Two important special classes of infinite abelian groups with diametrically opposite properties are torsion groups and torsion-free groups, examplified by the groups Q/Z (periodic) and Q (torsion-free).

Torsion groups An abelian group is called periodic or torsion if every element has finite order. A direct sum of finite cyclic groups is periodic. Although the converse statement is not true in general, some special cases are known. The first and second Prüfer theorems state that if A is a periodic group and either it has bounded exponent, i.e. nA = 0 for some natural number n, or if A is countable and the p-heights of the elements of A are finite for each p, then A is isomorphic to a direct sum of finite cyclic groups.[3] The cardinality of the set of direct summands isomorphic to Z/pmZ in such a decomposition is an invariant of A. These theorems were later subsumed in the Kulikov criterion. In a different direction, Helmut Ulm found an extension of the second Prüfer theorem to countable abelian p-groups with elements of infinite height: those groups are completely classified by means of their Ulm invariants.

Torsion-free and mixed groups An abelian group is called torsion-free if every non-zero element has infinite order. Several classes of torsion-free abelian groups have been extensively studied: • Free abelian groups, i.e. arbitrary direct sums of Z • Cotorsion and algebraically compact torsion-free groups such as the p-adic integers • Slender groups An abelian group that is neither periodic nor torsion-free is called mixed. If A is an abelian group and T(A) is its torsion subgroup then the factor group A/T(A) is torsion-free. However, in general the torsion subgroup is not a direct summand of A, so the torsion-free factor cannot be realized as a subgroup of A and A is not isomorphic to T(A) ⊕ A/T(A). Thus the theory of mixed groups involves more than simply combining the results about periodic and torsion-free groups.

Invariants and classification One of the most basic invariants of an infinite abelian group A is its rank: the cardinality of the maximal linearly independent subset of A. Abelian groups of rank 0 are precisely the periodic groups, while torsion-free abelian groups of rank 1 are necessarily subgroups of Q and can be completely described. More generally, a torsion-free abelian group of finite rank r is a subgroup of Qr. On the other hand, the group of p-adic integers Zp is a torsion-free abelian group of infinite Z-rank and the groups Zpn with different n are non-isomorphic, so this invariant does not even fully capture properties of some familiar groups.

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Abelian group The classification theorems for finitely generated, divisible, countable periodic, and rank 1 torsion-free abelian groups explained above were all obtained before 1950 and form a foundation of the classification of more general infinite abelian groups. Important technical tools used in classification of infinite abelian groups are pure and basic subgroups. Introduction of various invariants of torsion-free abelian groups has been one avenue of further progress. See the books by Irving Kaplansky, László Fuchs, Phillip Griffiths, and David Arnold, as well as the proceedings of the conferences on Abelian Group Theory published in Lecture Notes in Mathematics for more recent results.

Additive groups of rings The additive group of a ring is an abelian group, but not all abelian groups are additive groups of rings. Some important topics in this area of study are: • Tensor product • Corner's results on countable torsion-free groups • Shelah's work to remove cardinality restrictions

Relation to other mathematical topics Many large abelian groups possess a natural topology, which turns them into topological groups. The collection of all abelian groups, together with the homomorphisms between them, forms the category Ab, the prototype of an abelian category. Nearly all well-known algebraic structures other than Boolean algebras, are undecidable. Hence it is surprising that Tarski's student Szmielew (1955) proved that the first order theory of abelian groups, unlike its nonabelian counterpart, is decidable. This decidability, plus the fundamental theorem of finite abelian groups described above, highlight some of the successes in abelian group theory, but there are still many areas of current research: • Amongst torsion-free abelian groups of finite rank, only the finitely generated case and the rank 1 case are well understood; • There are many unsolved problems in the theory of infinite-rank torsion-free abelian groups; • While countable torsion abelian groups are well understood through simple presentations and Ulm invariants, the case of countable mixed groups is much less mature. • Many mild extensions of the first order theory of abelian groups are known to be undecidable. • Finite abelian groups remain a topic of research in computational group theory. Moreover, abelian groups of infinite order lead, quite surprisingly, to deep questions about the set theory commonly assumed to underlie all of mathematics. Take the Whitehead problem: are all Whitehead groups of infinite order also free abelian groups? In the 1970s, Saharon Shelah proved that the Whitehead problem is: • Undecidable in ZFC, the conventional axiomatic set theory from which nearly all of present day mathematics can be derived. The Whitehead problem is also the first question in ordinary mathematics proved undecidable in ZFC; • Undecidable even if ZFC is augmented by taking the generalized continuum hypothesis as an axiom; • Decidable if ZFC is augmented with the axiom of constructibility (see statements true in L).

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A note on the typography Among mathematical adjectives derived from the proper name of a mathematician, the word "abelian" is rare in that it is often spelled with a lowercase a, rather than an uppercase A, indicating how ubiquitous the concept is in modern mathematics.[4]

See also • • • • • • •

Abelianization Class field theory Commutator subgroup Elementary abelian group Pontryagin duality Pure injective module Pure projective module

Notes [1] Jacobson (2009), p. 41 [2] For example, Q/Z ≅ ∑p Qp/Zp. [3] Countability assumption in the second Prüfer theorem cannot be removed: the torsion subgroup of the direct product of the cyclic groups Z/pmZ for all natural m is not a direct sum of cyclic groups. [4] Abel Prize Awarded: The Mathematicians' Nobel (http:/ / www. maa. org/ devlin/ devlin_04_04. html)

References • Cox, David (2004) Galois Theory. Wiley-Interscience. Hoboken, NJ. xx+559 pp. MR2119052 • Fuchs, László (1970) Infinite abelian groups, Vol. I. Pure and Applied Mathematics, Vol. 36. New York–London: Academic Press. xi+290 pp. MR0255673 • ------ (1973) Infinite abelian groups, Vol. II. Pure and Applied Mathematics. Vol. 36-II. New York–London: Academic Press. ix+363 pp. MR0349869 • Griffith, Phillip A. (1970). Infinite Abelian group theory. Chicago Lectures in Mathematics. University of Chicago Press. ISBN 0-226-30870-7. • I.N. Herstein (1975), Topics in Algebra, 2nd edition (John Wiley and Sons, New York) ISBN 0-471-02371-X • Hillar, Christopher and Rhea, Darren (2007), Automorphisms of finite abelian groups. Amer. Math. Monthly 114, no. 10, 917-923. arXiv:0605185. • Jacobson, Nathan (2009). Basic algebra. 1 (2nd ed.). Dover. ISBN 978-0-486-47189-1.. • Szmielew, Wanda (1955) "Elementary properties of abelian groups," Fundamenta Mathematica 41: 203-71.

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Lie group In mathematics, a Lie group (pronounced /ˈliː/: similar to "Lee") is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. Lie groups are named after Sophus Lie, who laid the foundations of the theory of continuous transformation groups. Lie groups represent the best-developed theory of continuous symmetry of mathematical objects and structures, which makes them indispensable tools for many parts of contemporary mathematics, as well as for modern theoretical physics. They provide a natural framework for analysing the continuous symmetries of differential equations (Differential Galois theory), in much the same way as permutation groups are used in Galois theory for analysing the discrete symmetries of algebraic equations. An extension of Galois theory to the case of continuous symmetry groups was one of Lie's principal motivations.

Overview Lie groups are smooth manifolds and, therefore, can be studied using differential calculus, in contrast with the case of more general topological groups. One of the key ideas in the theory of Lie groups, from Sophus Lie, is to replace the global object, the group, with its local or linearized version, which Lie himself called its "infinitesimal group" and which has since become known as its Lie algebra. Lie groups play an enormous role in modern geometry, on several different levels. Felix Klein argued in his Erlangen program that one can consider various "geometries" by specifying an appropriate transformation group that leaves certain geometric properties invariant. Thus Euclidean geometry corresponds to the choice of the group E(3) The circle of center 0 and radius 1 in the complex of distance-preserving transformations of the Euclidean space R3, plane is a Lie group with complex multiplication. conformal geometry corresponds to enlarging the group to the conformal group, whereas in projective geometry one is interested in the properties invariant under the projective group. This idea later led to the notion of a G-structure, where G is a Lie group of "local" symmetries of a manifold. On a "global" level, whenever a Lie group acts on a geometric object, such as a Riemannian or a symplectic manifold, this action provides a measure of rigidity and yields a rich algebraic structure. The presence of continuous symmetries expressed via a Lie group action on a manifold places strong constraints on its geometry and facilitates analysis on the manifold. Linear actions of Lie groups are especially important, and are studied in representation theory. In the 1940s–1950s, Ellis Kolchin, Armand Borel and Claude Chevalley realised that many foundational results concerning Lie groups can be developed completely algebraically, giving rise to the theory of algebraic groups defined over an arbitrary field. This insight opened new possibilities in pure algebra, by providing a uniform construction for most finite simple groups, as well as in algebraic geometry. The theory of automorphic forms, an important branch of modern number theory, deals extensively with analogues of Lie groups over adele rings; p-adic Lie groups play an important role, via their connections with Galois representations in number theory.

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Definitions and examples A real Lie group is a group which is also a finite-dimensional real smooth manifold, and in which the group operations of multiplication and inversion are smooth maps. Smoothness of the group multiplication

means that μ is a smooth mapping of the product manifold G×G into G. These two requirements can be combined to the single requirement that the mapping

be a smooth mapping of the product manifold into G.

First examples • The 2×2 real invertible matrices form a group under multiplication, denoted by GL2(R):

This is a four-dimensional noncompact real Lie group. This group is disconnected; it has two connected components corresponding to the positive and negative values of the determinant. • The rotation matrices form a subgroup of GL2(R), denoted by SO2(R). It is a Lie group in its own right: specifically, a one-dimensional compact connected Lie group which is diffeomorphic to the circle. Using the rotation angle as a parameter, this group can be parametrized as follows:

Addition of the angles corresponds to multiplication of the elements of SO2(R), and taking the opposite angle corresponds to inversion. Thus both multiplication and inversion are differentiable maps. • The orthogonal group also forms an interesting example of a Lie group. All of the previous examples of Lie groups fall within the class of classical groups

Related concepts A complex Lie group is defined in the same way using complex manifolds rather than real ones (example: SL2(C)), and similarly one can define a p-adic Lie group over the p-adic numbers. Hilbert's fifth problem asked whether replacing differentiable manifolds with topological or analytic ones can yield new examples. The answer to this question turned out to be negative: in 1952, Gleason, Montgomery and Zippin showed that if G is a topological manifold with continuous group operations, then there exists exactly one analytic structure on G which turns it into a Lie group (see also Hilbert–Smith conjecture). If the underlying manifold is allowed to be infinite dimensional (for example, a Hilbert manifold) then one arrives at the notion of an infinite-dimensional Lie group. It is possible to define analogues of many Lie groups over finite fields, and these give most of the examples of finite simple groups. The language of category theory provides a concise definition for Lie groups: a Lie group is a group object in the category of smooth manifolds. This is important, because it allows generalization of the notion of a Lie group to Lie supergroups.

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More examples of Lie groups Lie groups occur in abundance throughout mathematics and physics. Matrix groups or algebraic groups are (roughly) groups of matrices (for example, orthogonal and symplectic groups), and these give most of the more common examples of Lie groups.

Examples • Euclidean space Rn with ordinary vector addition as the group operation becomes an n-dimensional noncompact abelian Lie group. • The circle group S1 consisting of angles mod 2π under addition or, alternately, the complex numbers with absolute value 1 under multiplication. This is a one-dimensional compact connected abelian Lie group. • The group GLn(R) of invertible matrices (under matrix multiplication) is a Lie group of dimension n2, called the general linear group. It has a closed connected subgroup SLn(R), the special linear group, consisting of matrices of determinant 1 which is also a Lie group. • The orthogonal group On(R), consisting of all n × n orthogonal matrices with real entries is an n(n − 1)/2-dimensional Lie group. This group is disconnected, but it has a connected subgroup SOn(R) of the same dimension consisting of orthogonal matrices of determinant 1, called the special orthogonal group (for n = 3, the rotation group). • The Euclidean group En(R) is the Lie group of all Euclidean motions, i.e., isometric affine maps, of n-dimensional Euclidean space Rn. • The unitary group U(n) consisting of n × n unitary matrices (with complex entries) is a compact connected Lie group of dimension n2. Unitary matrices of determinant 1 form a closed connected subgroup of dimension n2 − 1 denoted SU(n), the special unitary group. • Spin groups are double covers of the special orthogonal groups, used for studying fermions in quantum field theory (among other things). • The symplectic group Sp2n(R) consists of all 2n × 2n matrices preserving a nondegenerate skew-symmetric bilinear form on R2n (the symplectic form). It is a connected Lie group of dimension 2n2 + n. The fundamental group of the symplectic group is Z and this fact is related to the theory of Maslov index. • The 3-sphere S3 forms a Lie group by identification with the set of quaternions of unit norm, called versors. The only other spheres that admit the structure of a Lie group are the 0-sphere S0 (real numbers with absolute value 1) and the circle S1 (complex numbers with absolute value 1). For example, for even n > 1, Sn is not a Lie group because it does not admit a nonvanishing vector field and so a fortiori cannot be parallelizable as a differentiable manifold. Of the spheres only S0, S1, S3, and S7 are parallelizable. The latter carries the structure of a Lie quasigroup (a nonassociative group), which can be identified with the set of unit octonions. • The group of upper triangular n by n matrices is a solvable Lie group of dimension n(n + 1)/2. • The Lorentz group and the Poincare group are the groups of linear and affine isometries of the Minkowski space (interpreted as the spacetime of the special relativity). They are Lie groups of dimensions 6 and 10. • The Heisenberg group is a connected nilpotent Lie group of dimension 3, playing a key role in quantum mechanics. • The group U(1)×SU(2)×SU(3) is a Lie group of dimension 1+3+8=12 that is the gauge group of the Standard Model in particle physics. The dimensions of the factors correspond to the 1 photon + 3 vector bosons + 8 gluons of the standard model. • The (3-dimensional) metaplectic group is a double cover of SL2(R) playing an important role in the theory of modular forms. It is a connected Lie group that cannot be faithfully represented by matrices of finite size, i.e., a nonlinear group. • The exceptional Lie groups of types G2, F4, E6, E7, E8 have dimensions 14, 52, 78, 133, and 248. There is also a group E7½ of dimension 190.

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Constructions There are several standard ways to form new Lie groups from old ones: • • • •

The product of two Lie groups is a Lie group. Any topologically closed subgroup of a Lie group is a Lie group. This is known as Cartan's theorem. The quotient of a Lie group by a closed normal subgroup is a Lie group. The universal cover of a connected Lie group is a Lie group. For example, the group R is the universal cover of the circle group S1. In fact any covering of a differentiable manifold is also a differentiable manifold, but by specifying universal cover, one guarantees a group structure (compatible with its other structures).

Related notions Some examples of groups that are not Lie groups (except in the trivial sense that any group can be viewed as a 0-dimensional Lie group, with the discrete topology), are: • Infinite dimensional groups, such as the additive group of an infinite dimensional real vector space. These are not Lie groups as they are not finite dimensional manifolds • Some totally disconnected groups, such as the Galois group of an infinite extension of fields, or the additive group of the p-adic numbers. These are not Lie groups because their underlying spaces are not real manifolds. (Some of these groups are "p-adic Lie groups"). In general, only topological groups having similar local properties to Rn for some positive integer n can be Lie groups (of course they must also have a differentiable structure)

Early history According to the most authoritative source on the early history of Lie groups (Hawkins, p. 1), Sophus Lie himself considered the winter of 1873–1874 as the birth date of his theory of continuous groups. Hawkins, however, suggests that it was "Lie's prodigious research activity during the four-year period from the fall of 1869 to the fall of 1873" that led to the theory's creation (ibid). Some of Lie's early ideas were developed in close collaboration with Felix Klein. Lie met with Klein every day from October 1869 through 1872: in Berlin from the end of October 1869 to the end of February 1870, and in Paris, Göttingen and Erlangen in the subsequent two years (ibid, p. 2). Lie stated that all of the principal results were obtained by 1884. But during the 1870s all his papers (except the very first note) were published in Norwegian journals, which impeded recognition of the work throughout the rest of Europe (ibid, p. 76). In 1884 a young German mathematician, Friedrich Engel, came to work with Lie on a systematic treatise to expose his theory of continuous groups. From this effort resulted the three-volume Theorie der Transformationsgruppen, published in 1888, 1890, and 1893. Lie's ideas did not stand in isolation from the rest of mathematics. In fact, his interest in the geometry of differential equations was first motivated by the work of Carl Gustav Jacobi, on the theory of partial differential equations of first order and on the equations of classical mechanics. Much of Jacobi's work was published posthumously in the 1860s, generating enormous interest in France and Germany (Hawkins, p. 43). Lie's idée fixe was to develop a theory of symmetries of differential equations that would accomplish for them what Évariste Galois had done for algebraic equations: namely, to classify them in terms of group theory. Lie and other mathematicians showed that the most important equations for special functions and orthogonal polynomials tend to arise from group theoretical symmetries. Additional impetus to consider continuous groups came from ideas of Bernhard Riemann, on the foundations of geometry, and their further development in the hands of Klein. Thus three major themes in 19th century mathematics were combined by Lie in creating his new theory: the idea of symmetry, as exemplified by Galois through the algebraic notion of a group; geometric theory and the explicit solutions of differential equations of mechanics, worked out by Poisson and Jacobi; and the new understanding of geometry that emerged in the works of Plücker, Möbius, Grassmann and others, and culminated in Riemann's revolutionary vision of the subject.

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Lie group Although today Sophus Lie is rightfully recognized as the creator of the theory of continuous groups, a major stride in the development of their structure theory, which was to have a profound influence on subsequent development of mathematics, was made by Wilhelm Killing, who in 1888 published the first paper in a series entitled Die Zusammensetzung der stetigen endlichen Transformationsgruppen (The composition of continuous finite transformation groups) (Hawkins, p. 100). The work of Killing, later refined and generalized by Élie Cartan, led to classification of semisimple Lie algebras, Cartan's theory of symmetric spaces, and Hermann Weyl's description of representations of compact and semisimple Lie groups using highest weights. Weyl brought the early period of the development of the theory of Lie groups to fruition, for not only did he classify irreducible representations of semisimple Lie groups and connect the theory of groups with quantum mechanics, but he also put Lie's theory itself on firmer footing by clearly enunciating the distinction between Lie's infinitesimal groups (i.e., Lie algebras) and the Lie groups proper, and began investigations of topology of Lie groups (Borel (2001), ). The theory of Lie groups was systematically reworked in modern mathematical language in a monograph by Claude Chevalley.

The concept of a Lie group, and possibilities of classification Lie groups may be thought of as smoothly varying families of symmetries. Examples of symmetries include rotation about an axis. What must be understood is the nature of 'small' transformations, e.g., rotations through tiny angles, that link nearby transformations. The mathematical object capturing this structure is called a Lie algebra (Lie himself called them "infinitesimal groups"). It can be defined because Lie groups are manifolds, so have tangent spaces at each point. The Lie algebra of any compact Lie group (very roughly: one for which the symmetries form a bounded set) can be decomposed as a direct sum of an abelian Lie algebra and some number of simple ones. The structure of an abelian Lie algebra is mathematically uninteresting (since the Lie bracket is identically zero); the interest is in the simple summands. Hence the question arises: what are the simple Lie algebras of compact groups? It turns out that they mostly fall into four infinite families, the "classical Lie algebras" An, Bn, Cn and Dn, which have simple descriptions in terms of symmetries of Euclidean space. But there are also just five "exceptional Lie algebras" that do not fall into any of these families. E8 is the largest of these.

Properties • The diffeomorphism group of a Lie group acts transitively on the Lie group • Every Lie group is parallelizable, and hence an orientable manifold (there is a bundle isomorphism between its tangent bundle and the product of itself with the tangent space at the identity)

Types of Lie groups and structure theory Lie groups are classified according to their algebraic properties (simple, semisimple, solvable, nilpotent, abelian), their connectedness (connected or simply connected) and their compactness. • Compact Lie groups are all known: they are finite central quotients of a product of copies of the circle group S1 and simple compact Lie groups (which correspond to connected Dynkin diagrams). • Any simply connected solvable Lie group is isomorphic to a closed subgroup of the group of invertible upper triangular matrices of some rank, and any finite dimensional irreducible representation of such a group is 1 dimensional. Solvable groups are too messy to classify except in a few small dimensions. • Any simply connected nilpotent Lie group is isomorphic to a closed subgroup of the group of invertible upper triangular matrices with 1's on the diagonal of some rank, and any finite dimensional irreducible representation of such a group is 1 dimensional. Like solvable groups, nilpotent groups are too messy to classify except in a few small dimensions.

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• Simple Lie groups are sometimes defined to be those that are simple as abstract groups, and sometimes defined to be connected Lie groups with a simple Lie algebra. For example, SL2(R) is simple according to the second definition but not according to the first. They have all been classified (for either definition). • Semisimple Lie groups are Lie groups whose Lie algebra is a product of simple Lie algebras.[1] They are central extensions of products of simple Lie groups. The identity component of any Lie group is an open normal subgroup, and the quotient group is a discrete group. The universal cover of any connected Lie group is a simply connected Lie group, and conversely any connected Lie group is a quotient of a simply connected Lie group by a discrete normal subgroup of the center. Any Lie group G can be decomposed into discrete, simple, and abelian groups in a canonical way as follows. Write Gcon for the connected component of the identity Gsol for the largest connected normal solvable subgroup Gnil for the largest connected normal nilpotent subgroup so that we have a sequence of normal subgroups 1 ⊆ Gnil ⊆ Gsol ⊆ Gcon ⊆ G. Then G/Gcon is discrete Gcon/Gsol is a central extension of a product of simple connected Lie groups. Gsol/Gnil is abelian. A connected abelian Lie group is isomorphic to a product of copies of R and the circle group S1. Gnil/1 is nilpotent, and therefore its ascending central series has all quotients abelian. This can be used to reduce some problems about Lie groups (such as finding their unitary representations) to the same problems for connected simple groups and nilpotent and solvable subgroups of smaller dimension.

The Lie algebra associated with a Lie group To every Lie group, we can associate a Lie algebra, whose underlying vector space is the tangent space of G at the identity element, which completely captures the local structure of the group. Informally we can think of elements of the Lie algebra as elements of the group that are "infinitesimally close" to the identity, and the Lie bracket is something to do with the commutator of two such infinitesimal elements. Before giving the abstract definition we give a few examples: • The Lie algebra of the vector space Rn is just Rn with the Lie bracket given by [A, B] = 0. (In general the Lie bracket of a connected Lie group is always 0 if and only if the Lie group is abelian.) • The Lie algebra of the general linear group GLn(R) of invertible matrices is the vector space Mn(R) of square matrices with the Lie bracket given by [A, B] = AB − BA. If G is a closed subgroup of GLn(R) then the Lie algebra of G can be thought of informally as the matrices m of Mn(R) such that 1 + εm is in G, where ε is an infinitesimal positive number with ε2 = 0 (of course, no such real number ε exists). For example, the orthogonal group On(R) consists of matrices A with AAT = 1, so the Lie algebra consists of the matrices m with (1 + εm)(1 + εm)T = 1, which is equivalent to m + mT = 0 because ε2 = 0. • Formally, when working over the reals, as here, this is accomplished by considering the limit as ε → 0; but the "infinitesimal" language generalizes directly to Lie groups over general rings. The concrete definition given above is easy to work with, but has some minor problems: to use it we first need to represent a Lie group as a group of matrices, but not all Lie groups can be represented in this way, and it is not

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obvious that the Lie algebra is independent of the representation we use. To get round these problems we give the general definition of the Lie algebra of any Lie group (in 4 steps): 1. Vector fields on any smooth manifold M can be thought of as derivations X of the ring of smooth functions on the manifold, and therefore form a Lie algebra under the Lie bracket [X, Y] = XY − YX, because the Lie bracket of any two derivations is a derivation. 2. If G is any group acting smoothly on the manifold M, then it acts on the vector fields, and the vector space of vector fields fixed by the group is closed under the Lie bracket and therefore also forms a Lie algebra. 3. We apply this construction to the case when the manifold M is the underlying space of a Lie group G, with G acting on G = M by left translations Lg(h) = gh. This shows that the space of left invariant vector fields (vector fields satisfying Lg*Xh = Xgh for every h in G, where Lg* denotes the differential of Lg) on a Lie group is a Lie algebra under the Lie bracket of vector fields. 4. Any tangent vector at the identity of a Lie group can be extended to a left invariant vector field by left translating the tangent vector to other points of the manifold. Specifically, the left invariant extension of an element v of the tangent space at the identity is the vector field defined by v^g = Lg*v. This identifies the tangent space Te at the identity with the space of left invariant vector fields, and therefore makes the tangent space at the identity into a Lie algebra, called the Lie algebra of G, usually denoted by a Fraktur Thus the Lie bracket on is given explicitly by [v, w] = [v^, w^]e. This Lie algebra is finite-dimensional and it has the same dimension as the manifold G. The Lie algebra of G determines G up to "local isomorphism", where two Lie groups are called locally isomorphic if they look the same near the identity element. Problems about Lie groups are often solved by first solving the corresponding problem for the Lie algebras, and the result for groups then usually follows easily. For example, simple Lie groups are usually classified by first classifying the corresponding Lie algebras. We could also define a Lie algebra structure on Te using right invariant vector fields instead of left invariant vector fields. This leads to the same Lie algebra, because the inverse map on G can be used to identify left invariant vector fields with right invariant vector fields, and acts as −1 on the tangent space Te. The Lie algebra structure on Te can also be described as follows: the commutator operation (x, y) → xyx−1y−1 on G × G sends (e, e) to e, so its derivative yields a bilinear operation on TeG. This bilinear operation is actually the zero map, but the second derivative, under the proper identification of tangent spaces, yields an operation that satisfies the axioms of a Lie bracket, and it is equal to twice the one defined through left-invariant vector fields.

Homomorphisms and isomorphisms If G and H are Lie groups, then a Lie-group homomorphism f : G → H is a smooth group homomorphism. (It is equivalent to require only that f be continuous rather than smooth.) The composition of two such homomorphisms is again a homomorphism, and the class of all Lie groups, together with these morphisms, forms a category. Two Lie groups are called isomorphic if there exists a bijective homomorphism between them whose inverse is also a homomorphism. Isomorphic Lie groups are essentially the same; they only differ in the notation for their elements. Every homomorphism f : G → H of Lie groups induces a homomorphism between the corresponding Lie algebras and . The association G is a functor (mapping between categories satisfying certain axioms). One version of Ado's theorem is that every finite dimensional Lie algebra is isomorphic to a matrix Lie algebra. For every finite dimensional matrix Lie algebra, there is a linear group (matrix Lie group) with this algebra as its Lie algebra. So every abstract Lie algebra is the Lie algebra of some (linear) Lie group. The global structure of a Lie group is not determined by its Lie algebra; for example, if Z is any discrete subgroup of the center of G then G and G/Z have the same Lie algebra (see the table of Lie groups for examples). A connected Lie group is simple, semisimple, solvable, nilpotent, or abelian if and only if its Lie algebra has the corresponding

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property. If we require that the Lie group be simply connected, then the global structure is determined by its Lie algebra: for every finite dimensional Lie algebra over F there is a simply connected Lie group G with as Lie algebra, unique up to isomorphism. Moreover every homomorphism between Lie algebras lifts to a unique homomorphism between the corresponding simply connected Lie groups.

The exponential map The exponential map from the Lie algebra Mn(R) of the general linear group GLn(R) to GLn(R) is defined by the usual power series:

for matrices A. If G is any subgroup of GLn(R), then the exponential map takes the Lie algebra of G into G, so we have an exponential map for all matrix groups. The definition above is easy to use, but it is not defined for Lie groups that are not matrix groups, and it is not clear that the exponential map of a Lie group does not depend on its representation as a matrix group. We can solve both problems using a more abstract definition of the exponential map that works for all Lie groups, as follows. Every vector v in determines a linear map from R to taking 1 to v, which can be thought of as a Lie algebra homomorphism. Because R is the Lie algebra of the simply connected Lie group R, this induces a Lie group homomorphism c : R → G so that

for all s and t. The operation on the right hand side is the group multiplication in G. The formal similarity of this formula with the one valid for the exponential function justifies the definition

This is called the exponential map, and it maps the Lie algebra into the Lie group G. It provides a diffeomorphism between a neighborhood of 0 in and a neighborhood of e in G. This exponential map is a generalization of the exponential function for real numbers (because R is the Lie algebra of the Lie group of positive real numbers with multiplication), for complex numbers (because C is the Lie algebra of the Lie group of non-zero complex numbers with multiplication) and for matrices (because Mn(R) with the regular commutator is the Lie algebra of the Lie group GLn(R) of all invertible matrices). Because the exponential map is surjective on some neighbourhood N of e, it is common to call elements of the Lie algebra infinitesimal generators of the group G. The subgroup of G generated by N is the identity component of G. The exponential map and the Lie algebra determine the local group structure of every connected Lie group, because of the Baker–Campbell–Hausdorff formula: there exists a neighborhood U of the zero element of , such that for u, v in U we have exp(u) exp(v) = exp(u + v + 1/2 [u, v] + 1/12 [[u, v], v] − 1/12 [[u, v], u] − ...) where the omitted terms are known and involve Lie brackets of four or more elements. In case u and v commute, this formula reduces to the familiar exponential law exp(u) exp(v) = exp(u + v). The exponential map from the Lie algebra to the Lie group is not always onto, even if the group is connected (though it does map onto the Lie group for connected groups that are either compact or nilpotent). For example, the exponential map of SL2(R) is not surjective.

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Infinite dimensional Lie groups Lie groups are often defined to be finite dimensional, but there are many groups that resemble Lie groups, except for being infinite dimensional. The simplest was to define infinite dimensional Lie groups is to model them on Banach spaces, and in this case much of the basic theory is similar to that of finite dimensional lie groups. However this is inadequate for many applications, because many natural examples of infinite dimensional Lie groups are not Banach manifolds. Instead one needs to define Lie groups modeled on more general locally convex topological vector spaces. In this case the relation between the Lie algebra and the Lie group becomes rather subtle, and several results about finite dimensional Lie groups no longer hold. Some of the examples that have been studied include: • The group of diffeomorphisms of a manifold. Quite a lot is known about the group of diffeomorphisms of the circle. Its Lie algebra is (more or less) the Witt algebra, which has a central extension called the Virasoro algebra, used in string theory and conformal field theory. Very little is known about the diffeomorphism groups of manifolds of larger dimension. The diffeomorphism group of spacetime sometimes appears in attempts to quantize gravity. • The group of smooth maps from a manifold to a finite dimensional Lie group is an example of a gauge group (with operation of pointwise multiplication), and is used in quantum field theory and Donaldson theory. If the manifold is a circle these are called loop groups, and have central extensions whose Lie algebras are (more or less) Kac–Moody algebras. • There are infinite dimensional analogues of general linear groups, orthogonal groups, and so on. One important aspect is that these may have simpler topological properties: see for example Kuiper's theorem.

Notes [1] Sigurdur Helgason, "Differential Geometry, Lie Groups, and Symmetric Spaces", Academic Press, 1978, page 131.

References • Adams, John Frank (1969), Lectures on Lie Groups, Chicago Lectures in Mathematics, Chicago: Univ. of Chicago Press, ISBN 0-226-00527-5. • Borel, Armand (2001), Essays in the history of Lie groups and algebraic groups (http://books.google.com/ books?isbn=0821802887), History of Mathematics, 21, Providence, R.I.: American Mathematical Society, MR1847105, ISBN 978-0-8218-0288-5 • Bourbaki, Nicolas, Elements of mathematics: Lie groups and Lie algebras. Chapters 1–3 ISBN 3-540-64242-0, Chapters 4–6 ISBN 3-540-42650-7, Chapters 7–9 ISBN 3-540-43405-4 • Chevalley, Claude (1946), Theory of Lie groups, Princeton: Princeton University Press, ISBN 0-691-04990-4. • Fulton, William; Harris, Joe (1991), Representation theory. A first course, Graduate Texts in Mathematics, Readings in Mathematics, 129, New York: Springer-Verlag, MR1153249, ISBN 978-0-387-97527-6, ISBN 978-0-387-97495-8 • Hall, Brian C. (2003), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Springer, ISBN 0-387-40122-9. • Hawkins, Thomas (2000), Emergence of the theory of Lie groups (http://books.google.com/ books?isbn=978-0-387-98963-1), Sources and Studies in the History of Mathematics and Physical Sciences, Berlin, New York: Springer-Verlag, MR1771134, ISBN 978-0-387-98963-1 Borel's review (http://www.jstor. org/stable/2695575) • Knapp, Anthony W. (2002), Lie Groups Beyond an Introduction, Progress in Mathematics, 140 (2nd ed.), Boston: Birkhäuser, ISBN 0-8176-4259-5. • Rossmann, Wulf (2001), Lie Groups: An Introduction Through Linear Groups, Oxford Graduate Texts in Mathematics, Oxford University Press, ISBN 978-0198596837. The 2003 reprint corrects several typographical

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Lie group mistakes. • Serre, Jean-Pierre (1965), Lie Algebras and Lie Groups: 1964 Lectures given at Harvard University, Lecture notes in mathematics, 1500, Springer, ISBN 3-540-55008-9. • Steeb, Willi-Hans (2007), Continuous Symmetries, Lie algebras, Differential Equations and Computer Algebra: second edition, World Scientific Publishing, ISBN 981-270-809-X.

Galois group In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions (and polynomials which give rise to them) via Galois groups is called Galois theory, so named in honor of Évariste Galois who first discovered them. For a more elementary discussion of Galois groups in terms of permutation groups, see the article on Galois theory.

Definition Suppose that E is an extension of the field F (written as E/F and read E over F). Consider the set of all automorphisms of E/F (that is, isomorphisms α from E to itself such that α(x) = x for every x in F). This set of automorphisms with the operation of function composition forms a group, sometimes denoted by Aut(E/F). If E/F is a Galois extension, then Aut(E/F) is called the Galois group of (the extension) E over F, and is usually denoted by Gal(E/F).[1]

Examples In the following examples F is a field, and C, R, Q are the fields of complex, real, and rational numbers, respectively. The notation F(a) indicates the field extension obtained by adjoining an element a to the field F. • Gal(F/F) is the trivial group that has a single element, namely the identity automorphism. • Gal(C/R) has two elements, the identity automorphism and the complex conjugation automorphism. • Aut(R/Q) is trivial. Indeed it can be shown that any Q-automorphism must preserve the ordering of the real numbers and hence must be the identity. • Aut(C/Q) is an infinite group. • Gal(Q(√2)/Q) has two elements, the identity automorphism and the automorphism which exchanges √2 and −√2. • Consider the field K = Q(³√2). The group Aut(K/Q) contains only the identity automorphism. This is because K is not a normal extension, since the other two cube roots of 2 (both complex) are missing from the extension — in other words K is not a splitting field. • Consider now L = Q(³√2, ω), where ω is a primitive third root of unity. The group Gal(L/Q) is isomorphic to S3, the dihedral group of order 6, and L is in fact the splitting field of x3 − 2 over Q. • If q is a prime power, and if F = GF(q) and E = GF(qn) denote the Galois fields of order q and qn respectively, then Gal(E/F) is cyclic of order n.

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Properties The significance of an extension being Galois is that it obeys the fundamental theorem of Galois theory: the closed (with respect to the Krull topology below) subgroups of the Galois group correspond to the intermediate fields of the field extension. If E/F is a Galois extension, then Gal(E/F) can be given a topology, called the Krull topology, that makes it into a profinite group.

Notes [1] Some authors refer to Aut(E/F) as the Galois group for arbitrary extensions E/F and use the corresponding notation, e.g. Jacobson 2009.

References • Jacobson, Nathan (2009) [1985], Basic algebra I (Second ed.), Dover Publications, ISBN 978-0-486-47189-1 • Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, 211 (Revised third ed.), New York: Springer-Verlag, MR1878556, ISBN 978-0-387-95385-4

External links • Galois Groups (http://www.mathpages.com/home/kmath290/kmath290.htm) at MathPages

General linear group In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible. The name is because the columns of an invertible matrix are linearly independent, hence the vectors/points they define are in general linear position, and matrices in the general linear group take points in general linear position to points in general linear position. To be more precise, it is necessary to specify what kind of objects may appear in the entries of the matrix. For example, the general linear group over R (the set of real numbers) is the group of n×n invertible matrices of real numbers, and is denoted by GLn(R) or GL(n, R). More generally, the general linear group of degree n over any field F (such as the complex numbers), or a ring R (such as the ring of integers), is the set of n×n invertible matrices with entries from F (or R), again with matrix multiplication as the group operation.[1] Typical notation is GLn(F) or GL(n, F), or simply GL(n) if the field is understood. More generally still, the general linear group of a vector space GL(V) is the abstract automorphism group, not necessarily written as matrices. The special linear group, written SL(n, F) or SLn(F), is the subgroup of GL(n, F) consisting of matrices with a determinant of 1. The group GL(n, F) and its subgroups are often called linear groups or matrix groups (the abstract group GL(V) is a linear group but not a matrix group). These groups are important in the theory of group representations, and also arise in the study of spatial symmetries and symmetries of vector spaces in general, as well as the study of polynomials. The modular group may be realised as a quotient of the special linear group SL(2, Z). If n ≥ 2, then the group GL(n, F) is not abelian.

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General linear group of a vector space If V is a vector space over the field F, the general linear group of V, written GL(V) or Aut(V), is the group of all automorphisms of V, i.e. the set of all bijective linear transformations V → V, together with functional composition as group operation. If V has finite dimension n, then GL(V) and GL(n, F) are isomorphic. The isomorphism is not canonical; it depends on a choice of basis in V. Given a basis (e1, ..., en) of V and an automorphism T in GL(V), we have

for some constants ajk in F; the matrix corresponding to T is then just the matrix with entries given by the ajk. In a similar way, for a commutative ring R the group GL(n, R) may be interpreted as the group of automorphisms of a free R-module M of rank n. One can also define GL(M) for any R-module, but in general this is not isomorphic to GL(n, R) (for any n).

In terms of determinants Over a field F, a matrix is invertible if and only if its determinant is nonzero. Therefore an alternative definition of GL(n, F) is as the group of matrices with nonzero determinant. Over a commutative ring R, one must be slightly more careful: a matrix over R is invertible if and only if its determinant is a unit in R, that is, if its determinant is invertible in R. Therefore GL(n, R) may be defined as the group of matrices whose determinants are units. Over a non-commutative ring R, determinants are not at all well behaved. In this case, GL(n, R) may be defined as the unit group of the matrix ring M(n, R).

As a Lie group Real case The general linear group GL(n,R) over the field of real numbers is a real Lie group of dimension n2. To see this, note that the set of all n×n real matrices, Mn(R), forms a real vector space of dimension n2. The subset GL(n,R) consists of those matrices whose determinant is non-zero. The determinant is a polynomial map, and hence GL(n,R) is a open affine subvariety of Mn(R) (a non-empty open subset of Mn(R) in the Zariski topology), and therefore[2] a smooth manifold of the same dimension. The Lie algebra of GL(n,R), denoted

consists of all n×n real matrices with the commutator serving as the Lie

bracket. As a manifold, GL(n,R) is not connected but rather has two connected components: the matrices with positive determinant and the ones with negative determinant. The identity component, denoted by GL+(n, R), consists of the real n×n matrices with positive determinant. This is also a Lie group of dimension n2; it has the same Lie algebra as GL(n,R). The group GL(n,R) is also noncompact. "The"[3] maximal compact subgroup of GL(n, R) is the orthogonal group O(n), while "the" maximal compact subgroup of GL+(n, R) is the special orthogonal group SO(n). As for SO(n), the group GL+(n, R) is not simply connected (except when n=1), but rather has a fundamental group isomorphic to Z for n=2 or Z2 for n>2.

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Complex case The general linear GL(n,C) over the field of complex numbers is a complex Lie group of complex dimension n2. As a real Lie group it has dimension 2n2. The set of all real matrices forms a real Lie subgroup. The Lie algebra corresponding to GL(n,C) consists of all n×n complex matrices with the commutator serving as the Lie bracket. Unlike the real case, GL(n,C) is connected. This follows, in part, since the multiplicative group of complex numbers C× is connected. The group manifold GL(n,C) is not compact; rather its maximal compact subgroup is the unitary group U(n). As for U(n), the group manifold GL(n,C) is not simply connected but has a fundamental group isomorphic to Z.

Over finite fields If F is a finite field with q elements, then we sometimes write GL(n, q) instead of GL(n, F). When p is prime, GL(n, p) is the outer automorphism group of the group Zpn, and also the automorphism group, because Zpn is Abelian, so the inner automorphism group is trivial. The order of GL(n, q) is: (qn − 1)(qn − q)(qn − q2) … (qn − qn−1) This can be shown by counting the possible columns of the matrix: the first column can be anything but the zero vector; the second column can be anything but the multiples of the first column; and in general, the kth column can be any vector not in the linear span of the first k − 1 columns. In q-analog notation, this is For example, GL(3, 2) has order (8 − 1)(8 − 2)(8 − 4) = 168. It is the automorphism group of the Fano plane and of the group Z23, and is also known as PSL(2,7). More generally, one can count points of Grassmannian over F: in other words the number of subspaces of a given dimension k. This requires only finding the order of the stabilizer subgroup of one such subspace (described on that page in block matrix form), and dividing into the formula just given, by the orbit-stabilizer theorem. These formulas are connected to the Schubert decomposition of the Grassmannian, and are q-analogs of the Betti numbers of complex Grassmannians. This was one of the clues leading to the Weil conjectures. Note that in the limit as

the order of GL(n, q) goes to

which is the order of the symmetric group – in the

philosophy of the field with one element, one thus interprets the symmetric group as the general linear group over the field with one element:

History The general linear group over a prime field, GL(ν,p), was constructed and its order computed by Évariste Galois in 1832, in his last letter (to Chevalier) and second (of three) attached manuscripts, which he used in the context of studying the Galois group of the general equation of order pν.[4]

Special linear group The special linear group, SL(n, F), is the group of all matrices with determinant 1. They are special in that they lie on a subvariety – they satisfy a polynomial equation (as the determinant is a polynomial in the entries). Matrices of this type form a group as the determinant of the product of two matrices is the product of the determinants of each matrix. SL(n, F) is a normal subgroup of GL(n, F). If we write F× for the multiplicative group of F (excluding 0), then the determinant is a group homomorphism det: GL(n, F) → F×.

General linear group The kernel of the map is just the special linear group. By the first isomorphism theorem we see that GL(n,F)/SL(n,F) is isomorphic to F×. In fact, GL(n, F) can be written as a semidirect product of SL(n, F) by F×: GL(n, F) = SL(n, F) ⋊ F× When F is R or C, SL(n) is a Lie subgroup of GL(n) of dimension n2 − 1. The Lie algebra of SL(n) consists of all n×n matrices over F with vanishing trace. The Lie bracket is given by the commutator. The special linear group SL(n, R) can be characterized as the group of volume and orientation preserving linear transformations of Rn. The group SL(n, C) is simply connected while SL(n, R) is not. SL(n, R) has the same fundamental group as GL+(n, R), that is, Z for n=2 and Z2 for n>2.

Other subgroups Diagonal subgroups The set of all invertible diagonal matrices forms a subgroup of GL(n, F) isomorphic to (F×)n. In fields like R and C, these correspond to rescaling the space; the so called dilations and contractions. A scalar matrix is a diagonal matrix which is a constant times the identity matrix. The set of all nonzero scalar matrices forms a subgroup of GL(n, F) isomorphic to F× . This group is the center of GL(n, F). In particular, it is a normal, abelian subgroup. The center of SL(n, F) is simply the set of all scalar matrices with unit determinant, and is isomorphic to the group of nth roots of unity in the field F.

Classical groups The so-called classical groups are subgroups of GL(V) which preserve some sort of bilinear form on a vector space V. These include the • orthogonal group, O(V), which preserves a non-degenerate quadratic form on V, • symplectic group, Sp(V), which preserves a symplectic form on V (a non-degenerate alternating form), • unitary group, U(V), which, when F = C, preserves a non-degenerate hermitian form on V. These groups provide important examples of Lie groups.

Related groups Projective linear group The projective linear group PGL(n, F) and the projective special linear group PSL(n,F) are the quotients of GL(n,F) and SL(n,F) by their centers (which consist of the multiples of the identity matrix therein); they are the induced action on the associated projective space.

Affine group The affine group Aff(n,F) is an extension of GL(n,F) by the group of translations in Fn. It can be written as a semidirect product: Aff(n, F) = GL(n, F) ⋉ Fn where GL(n, F) acts on Fn in the natural manner. The affine group can be viewed as the group of all affine transformations of the affine space underlying the vector space Fn. One has analogous constructions for other subgroups of the general linear group: for instance, the special affine group is the subgroup defined by the semidirect product, SL(n, F) ⋉ Fn, and the Poincaré group is the affine group

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General linear group associated to the Lorentz group, O(1,3,F) ⋉ Fn.

General semilinear group The general semilinear group ΓL(n,F) is the group of all invertible semilinear transformations, and contains GL. A semilinear transformation is a transformation which is linear "up to a twist", meaning "up to a field automorphism under scalar multiplication". It can be written as a semidirect product: ΓL(n, F) = Gal(F) ⋉ GL(n, F) where Gal(F) is the Galois group of F (over its prime field), which acts on GL(n, F) by the Galois action on the entries. The main interest of ΓL(n, F) is that the associated projective semilinear group PΓL(n, F) (which contains PGL(n, F)) is the collineation group of projective space, for n > 2, and thus semilinear maps are of interest in projective geometry.

Infinite general linear group The infinite general linear group or stable general linear group is the direct limit of the inclusions as the upper left block matrix. It is denoted by either or , and can also be interpreted as invertible infinite matrices which differ from the identity matrix in only finitely many places. It is used in algebraic K-theory to define K1, and over the reals has a well-understood topology, thanks to Bott periodicity. It should not be confused with the space of (bounded) invertible operators on a Hilbert space, which is a larger group, and topologically much simpler, namely contractible – see Kuiper's theorem.

See also • List of finite simple groups • SL2(R) • Representation theory of SL2(R)

Notes [1] [2] [3] [4]

Here rings are assumed to be associative and unital. Since the Zariski topology is coarser than the metric topology; equivalently, polynomial maps are continuous. A maximal compact subgroup is not unique, but is essentially unique, hence one often refers to "the" maximal compact subgroup. Galois, Évariste (1846). "Lettre de Galois à M. Auguste Chevalier" (http:/ / visualiseur. bnf. fr/ ark:/ 12148/ cb343487840/ date1846). Journal des mathématiques pures et appliquées XI: 408–415. . Retrieved 2009-02-04, GL(ν,p) discussed on p. 410.

External links • "GL(2,p) and GL(3,3) Acting on Points" (http://demonstrations.wolfram.com/GL2PAndGL33ActingOnPoints/ ) by Ed Pegg, Jr., Wolfram Demonstrations Project, 2007.

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Representation theory Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces.[1] In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and the algebraic operations in terms of matrix addition and matrix multiplication. The algebraic objects amenable to such a description include groups, associative algebras and Lie algebras. The most prominent of these (and historically the first) is the representation theory of groups, in which elements of a group are represented by invertible matrices in such a way that the group operation is matrix multiplication.[2] Representation theory is a powerful tool because it reduces problems in abstract algebra to problems in linear algebra, a subject which is well understood.[3] Furthermore, the vector space on which a group (for example) is represented can be infinite dimensional, and by allowing it to be, for instance, a Hilbert space, methods of analysis can be applied to the theory of groups.[4] Representation theory is also important in physics because, for example, it describes how the symmetry group of a physical system affects the solutions of equations describing that system.[5] A striking feature of representation theory is its pervasiveness in mathematics. There are two sides to this. First, the applications of representation theory are diverse:[6] in addition to its impact on algebra, representation theory illuminates and vastly generalizes Fourier analysis via harmonic analysis,[7] is deeply connected to geometry via invariant theory and the Erlangen program,[8] and has a profound impact in number theory via automorphic forms and the Langlands program.[9] The second aspect is the diversity of approaches to representation theory. The same objects can be studied using methods from algebraic geometry, module theory, analytic number theory, differential geometry, operator theory and topology.[10] The success of representation theory has led to numerous generalizations. One of the most general is a categorical one.[11] The algebraic objects to which representation theory applies can be viewed as particular kinds of categories, and the representations as functors from the object category to the category of vector spaces. This description points to two obvious generalizations: first, the algebraic objects can be replaced by more general categories; second the target category of vector spaces can be replaced by other well-understood categories.

Definitions and concepts Let V be a vector space over a field F.[3] For instance, suppose V is Rn or Cn, the standard n-dimensional space of column vectors over the real or complex numbers respectively. In this case, the idea of representation theory is to do abstract algebra concretely by using n × n matrices of real or complex numbers. There are three main sorts of algebraic objects for which this can be done: groups, associative algebras and Lie algebras.[12] • The set of all invertible n × n matrices is a group under matrix multiplication and the representation theory of groups analyses a group by describing ("representing") its elements in terms of invertible matrices. • Matrix addition and multiplication make the set of all n × n matrices into an associative algebra and hence there is a corresponding representation theory of associative algebras. • If we replace matrix multiplication MN by the matrix commutator MN − NM, then the n × n matrices become instead a Lie algebra, leading to a representation theory of Lie algebras. This generalizes to any field F and any vector space V over F, with linear maps replacing matrices and composition replacing matrix multiplication: there is a group GL(V,F) of automorphisms of V, an associative algebra EndF(V) of all endomorphisms of V, and a corresponding Lie algebra gl(V,F).

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Definition There are two ways to say what a representation is.[13] The first uses the idea of an action, generalizing the way that matrices act on column vectors by matrix multiplication. A representation of a group G or (associative or Lie) algebra A on a vector space V is a map

with two properties. First, for any g in G (or a in A), the map

is linear (over F), and similarly in the algebra cases. Second, if we introduce the notation g · v for Φ (g, v), then for any g1, g2 in G and v in V:

where e is the identity element of G and g1g2 is product in G. The requirement for associative algebras is analogous, except that associative algebras do not always have an identity element, in which case equation (1) is ignored. Equation (2) is an abstract expression of the associativity of matrix multiplication. This doesn't hold for the matrix commutator and also there is no identity element for the commutator. Hence for Lie algebras, the only requirement is that for any x1, x2 in A and v in V: where [x1, x2] is the Lie bracket, which generalizes the matrix commutator MN − NM. The second way to define a representation focuses on the map φ sending g in G to φ(g): V → V, which satisfies

and similarly in the other cases. This approach is both more concise and more abstract. • A representation of a group G on a vector space V is a group homomorphism φ: G → GL(V,F). • A representation of an associative algebra A on a vector space V is an algebra homomorphism φ: A → EndF(V). • A representation of a Lie algebra a on a vector space V is a Lie algebra homomorphism φ: a → gl(V,F).

Terminology The vector space V is called the representation space of φ and its dimension (if finite) is called the dimension of the representation. It is also common practice to refer to V itself as the representation when the homomorphism φ is clear from the context; otherwise the notation (V,φ) can be used to denote a representation. When V is of finite dimension n, one can choose a basis for V to identify V with Fn and hence recover a matrix representation with entries in the field F. An effective or faithful representation is a representation (V,φ) for which the homomorphism φ is injective.

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Equivariant maps and isomorphisms If V and W are vector spaces over F, equipped with representations φ and ψ of a group G, then an equivariant map from V to W is linear map α: V → W such that

for all g in G and v in V. In terms of φ: G → GL(V) and ψ: G → GL(W), this means

for all g in G. Equivariant maps for representations of an associative or Lie algebra are defined similarly. If α is invertible, then it is said to be an isomorphism, in which case V and W (or, more precisely, φ and ψ) are isomorphic representations. Isomorphic representations are, for all practical purposes, "the same": they provide the same information about the group or algebra being represented. Representation theory therefore seeks to classify representations "up to isomorphism".

Subrepresentations, quotients, and irreducible representations If (W,ψ) is a representation of (say) a group G, and V is a linear subspace of W which is preserved by the action of G in the sense that g · v ∈ V for all v ∈ V, then V is called a subrepresentation: by defining φ(g) to be the restriction of ψ(g) to V, (V, φ) is a representation of G and the inclusion of V into W is an equivariant map. The quotient space W/V can also be made into a representation of G. If W has exactly two subrepresentations, namely the trivial subspace {0} and W itself, then the representation is said to be irreducible; if W has a proper nontrivial subrepresentation, the representation is said to be reducible.[14] The definition of an irreducible representation implies Schur's lemma: an equivariant map α: V → irreducible representations is either the zero map or an isomorphism, since its kernel and subrepresentations. In particular, when V = W, this shows that the equivariant endomorphisms of associative division algebra over the underlying field F. If F is algebraically closed, the only endomorphisms of an irreducible representation are the scalar multiples of the identity.

W between image are V form an equivariant

Irreducible representations are the building blocks of representation theory: if a representation W is not irreducible then it is built from a subrepresentation and a quotient which are both "simpler" in some sense; for instance, if W is finite dimensional, then both the subrepresentation and the quotient have smaller dimension.

Direct sums and indecomposable representations If (V,φ) and (W,ψ) are representations of (say) a group G, then the direct sum of V and W is a representation, in a canonical way, via the equation

The direct sum of two representations carries no more information about the group G than the two representations do individually. If a representation is the direct sum of two proper nontrivial subrepresentations, it is said to be decomposable. Otherwise, it is said to be indecomposable. In favourable circumstances, every representation is a direct sum of irreducible representations: such representations are said to be semisimple. In this case, it suffices to understand only the irreducible representations. In other cases, one must understand how indecomposable representations can be built from irreducible representations as extensions of a quotient by a subrepresentation.

Representation theory

Branches and topics Representation theory is notable for the number of branches it has, and the diversity of the approaches to studying representations of groups and algebras. Although, all the theories have in common the basic concepts discussed already, they differ considerably in detail. The differences are at least 3-fold: 1. Representation theory depends upon the type of algebraic object being represented. There are several different classes of groups, associative algebras and Lie algebras, and their representation theories all have an individual flavour. 2. Representation theory depends upon the nature of the vector space on which the algebraic object is represented. The most important distinction is between finite dimensional representations and infinite dimensional ones. In the infinite-dimensional case, additional structures are important (e.g. whether or not the space is a Hilbert space, Banach space, etc.). Additional algebraic structures can also be imposed in the finite dimensional case. 3. Representation theory depends upon the type of field over which the vector space is defined. The most important case is the field of complex numbers. The other important cases are the field of real numbers, finite fields, and fields of p-adic numbers. Additional difficulties arise for fields of positive characteristic and for fields which are not algebraically closed.

Finite groups Group representations are a very important tool in the study of finite groups.[15] They also arise in the applications of finite group theory to geometry and crystallography.[16] Representations of finite groups exhibit many of the features of the general theory and point the way to other branches and topics in representation theory. Over a field of characteristic zero, the representation theory of a finite group G has a number of convenient properties. First, the representations of G are semisimple (completely reducible). This is a consequence of Maschke's theorem, which states that any subrepresentation V of a G-representation W has a G-invariant complement. One proof is to choose any projection π from W to V and replace it by its average πG defined by

πG is equivariant, and its kernel is the required complement. The finite dimensional G-representations can be understood using character theory: the character of a representation φ: G → GL(V) is the class function χφ: G → F defined by where

is the trace. An irreducible representation of G is completely determined by its character.

Maschke's theorem holds more generally for fields of positive characteristic p, such as the finite fields, as long as the prime p is coprime to the order of G. When p and |G| have a common factor, there are G-representations which are not semisimple, which are studied in a subbranch called modular representation theory. Averaging techniques also show that if F is the real or complex numbers, then any G-representation preserves an inner product on V in the sense that

for all g in G and v, w in W. Hence any G-representation is unitary. Unitary representations are automatically semisimple, since Maschke's result can be proven by taking the orthogonal complement of a subrepresentation. When studying representations of groups which are not finite, the unitary representations provide a good generalization of the real and complex representations of a finite group. Results such as Maschke's theorem and the unitary property which rely on averaging can be generalized to more general groups by replacing the average with an integral, provided that a suitable notion of integral can be defined. This can be done for compact groups or locally compact groups, using Haar measure, and the resulting theory is

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Representation theory known as abstract harmonic analysis. Over arbitrary fields, another class of finite groups which have a good representation theory are the finite groups of Lie type. Important examples are linear algebraic groups over finite fields. The representation theory of linear algebraic groups and Lie groups extends these examples to infinite dimensional groups, the latter being intimately related to Lie algebra representations. The importance of character theory for finite groups has an analogue in the theory of weights for representations of Lie groups and Lie algebras. Representations of a finite group G are also linked directly to algebra representations via the group algebra F[G], which is a vector space over F with the elements of G as a basis, equipped with the multiplication operation defined by the group operation, linearity, and the requirement that the group operation and scalar multiplication commute.

Modular representations Modular representations of a finite group G are representations over a field whose characteristic is not coprime to |G|, so that Maschke's theorem no longer holds (because |G| is not invertible in F and so one cannot divide by it).[17] Nevertheless, Richard Brauer extended much of character theory to modular representations, and this theory played an important role in early progress towards the classification of finite simple groups, especially for simple groups whose characterization was not amenable to purely group-theoretic methods because their Sylow 2-subgroups were "too small".[18] As well as having applications to group theory, modular representations arise naturally in other branches of mathematics, such as algebraic geometry, coding theory, combinatorics and number theory.

Unitary representations A unitary representation of a group G is a linear representation φ of G on a real or (usually) complex Hilbert space V such that φ(g) is a unitary operator for every g ∈ G. Such representations have been widely applied in quantum mechanics since the 1920s, thanks in particular to the influence of Hermann Weyl,[19] and this has inspired the development of the theory, most notably through the analysis of representations of the Poincare group by Eugene Wigner.[20] One of the pioneers in constructing a general theory of unitary representations (for any group G rather than just for particular groups useful in applications) was George Mackey, and an extensive theory was developed by Harish-Chandra and others in the 1950s and 1960s.[21] A major goal is to describe the "unitary dual", the space of irreducible unitary representations of G.[22] The theory is most well-developed in the case that G is a locally compact (Hausdorff) topological group and the representations are strongly continuous.[7] For G abelian, the unitary dual is just the space of characters, while for G compact, the Peter-Weyl theorem shows that the irreducible unitary representations are finite dimensional and the unitary dual is discrete.[23] For example, if G is the circle group S1, then the characters are given by integers, and the unitary dual is Z. For non-compact G, the question of which representations are unitary is a subtle one. Although irreducible unitary representations must be "admissible" (as Harish-Chandra modules) and it is easy to detect which admissible representations have a nondegenerate invariant sesquilinear form, it is hard to determine when this form is positive definite. An effective description of the unitary dual, even for relatively well-behaved groups such as real reductive Lie groups (discussed below), remains an important open problem in representation theory. It has been solved for many particular groups, such as SL(2,R) and the Lorentz group.[24]

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Harmonic analysis The duality between the circle group S1 and the integers Z, or more generally, between a torus Tn and Zn is well known in analysis as the theory of Fourier series, and the Fourier transform similarly expresses the fact that the space of characters on a real vector space is the dual vector space. Thus unitary representation theory and harmonic analysis are intimately related, and abstract harmonic analysis exploits this relationship, by developing the analysis of functions on locally compact topological groups and related spaces.[7] A major goal is to provide a general form of the Fourier transform and the Plancherel theorem. This is done by constructing a measure on the unitary dual and an isomorphism between the regular representation of G on the space L2(G) of square integrable functions on G and its representation on the space of L2 functions on the unitary dual. Pontrjagin duality and the Peter-Weyl theorem achieve this for abelian and compact G respectively.[23] [25] Another approach involves considering all unitary representations, not just the irreducible ones. These form a category, and Tannaka-Krein duality provides a way to recover a compact group from its category of unitary representations. If the group is neither abelian nor compact, no general theory is known with an analogue of the Plancherel theorem or Fourier inversion, although Alexander Grothendieck extended Tannaka-Krein duality to a relationship between linear algebraic groups and tannakian categories. Harmonic analysis has also been extended from the analysis of functions on a group G to functions on homogeneous spaces for G. The theory is particularly well developed for symmetric spaces and provides a theory of automorphic forms (discussed below).

Lie groups A Lie group is a group which is also a smooth manifold. Many classical groups of matrices over the real or complex numbers are Lie groups.[26] Many of the groups important in physics and chemistry are Lie groups, and their representation theory is crucial to the application of group theory in those fields.[5] The representation theory of Lie groups can be developed first by considering the compact groups, to which results of compact representation theory apply.[22] This theory can be extended to finite dimensional representations of semisimple Lie groups using Weyl's unitary trick: each semisimple real Lie group G has a complexification, which is a complex Lie group Gc, and this complex Lie group has a maximal compact subgroup K. The finite dimensional representations of G closely correspond to those of K. A general Lie group is a semidirect product of a solvable Lie group and a semisimple Lie group (the Levi decomposition).[27] The classification of representations of solvable Lie groups is intractable in general, but often easy in practical cases. Representations of semidirect products can then be analysed by means of general results called Mackey theory, which is a generalization of the methods used in Wigner's classification of representations of the Poincaré group.

Lie algebras A Lie algebra over a field F is a vector space over F equipped with a skew-symmetric bilinear operation called the Lie bracket, which satisfies the Jacobi identity. Lie algebras arise in particular as tangent spaces to Lie groups at the identity element, leading to their interpretation as "infinitesimal symmetries".[27] An important approach to the representation theory of Lie groups is to study the corresponding representation theory of Lie algebras, but representations of Lie algebras also have an intrinsic interest.[28] Lie algebras, like Lie groups, have a Levi decomposition into semisimple and solvable parts, with the representation theory of solvable Lie algebras being intractable in general. In contrast, the finite dimensional representations of semisimple Lie algebras are completely understood, after work of Elie Cartan. A representation of a semisimple Lie algebra g is analysed by choosing a Cartan subalgebra, which is essentially a generic maximal subalgebra h of g on

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Representation theory which the Lie bracket is zero ("abelian"). The representation of g can be decomposed into weight spaces which are eigenspaces for the action of h and the infinitesimal analogue of characters. The structure of semisimple Lie algebras then reduces the analysis of representations to easily understood combinatorics of the possible weights which can occur.[27] Infinite dimensional Lie algebras There are many classes of infinite dimensional Lie algebras whose representations have been studied. Among these, an important class are the Kac-Moody algebras.[29] They are named after Victor Kac and Robert Moody, who independently discovered them. These algebras form a generalization of finite-dimensional semisimple Lie algebras, and share many of their combinatorial properties. This means that they have a class of representations which can be understood in the same way as representations of semisimple Lie algebras. Affine Lie algebras are a special case of Kac-Moody algebras which have particular importance in mathematics and theoretical physics, especially conformal field theory and the theory of exactly solvable models. Kac discovered an elegant proof of certain combinatorial identities, Macdonald identities, which is based on the representation theory of affine Kac-Moody algebras. Lie superalgebras Lie superalgebras are generalizations of Lie algebras in which the underlying vector space has a Z2-grading, and skew-symmetry and Jacobi identity properties of the Lie bracket are modified by signs. Their representation theory is similar to the representation theory of Lie algebras.[30]

Linear algebraic groups Linear algebraic groups (or more generally, affine group schemes) are analogues in algebraic geometry of Lie groups, but over more general fields than just R or C. In particular, over finite fields, they give rise to finite groups of Lie type. Although linear algebraic groups have a classification that is very similar to that of Lie groups, their representation theory is rather different (and much less well understood) and requires different techniques, since the Zariski topology is relatively weak, and techniques from analysis are no longer available.[31]

Invariant theory Invariant theory studies actions on algebraic varieties from the point of view of their effect on functions, which form representations of the group. Classically, the theory dealt with the question of explicit description of polynomial functions that do not change, or are invariant, under the transformations from a given linear group. The modern approach analyses the decomposition of these representations into irreducibles.[32] Invariant theory of infinite groups is inextricably linked with the development of linear algebra, especially, the theories of quadratic forms and determinants. Another subject with strong mutual influence is projective geometry, where invariant theory can be used to organize the subject, and during the 1960s, new life was breathed into the subject by David Mumford in the form of his geometric invariant theory.[33] The representation theory of semisimple Lie groups has its roots in invariant theory[26] and the strong links between representation theory and algebraic geometry have many parallels in differential geometry, beginning with Felix Klein's Erlangen program and Elie Cartan's connections, which place groups and symmetry at the heart of geometry.[34] Modern developments link representation theory and invariant theory to areas as diverse as holonomy, differential operators and the theory of several complex variables.

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Automorphic forms and number theory Automorphic forms are a generalization of modular forms to more general analytic functions, perhaps of several complex variables, with similar transformation properties.[35] The generalization involves replacing the modular group PSL2 (R) and a chosen congruence subgroup by a semisimple Lie group G and a discrete subgroup Γ. Just as modular forms can be viewed as differential forms on a quotient of the upper half space H = PSL2 (R)/SO(2), automorphic forms can be viewed as differential forms (or similar objects) on Γ\G/K, where K is (typically) a maximal compact subgroup of G. Some care is required, however, as the quotient typically has singularities. The quotient of a semisimple Lie group by a compact subgroup is a symmetric space and so the theory of automorphic forms is intimately related to harmonic analysis on symmetric spaces. Before the development of the general theory, many important special cases were worked out in detail, including the Hilbert modular forms and Siegel modular forms. Important results in the theory include the Selberg trace formula and the realization by Robert Langlands that the Riemann-Roch theorem could be applied to calculate the dimension of the space of automorphic forms. The subsequent notion of "automorphic representation" has proved of great technical value for dealing with the case that G is an algebraic group, treated as an adelic algebraic group. As a result an entire philosophy, the Langlands program has developed around the relation between representation and number theoretic properties of automorphic forms.[36]

Associative algebras In one sense, associative algebra representations generalize both representations of groups and Lie algebras. A representation of a group induces a representation of a corresponding group ring or group algebra, while representations of a Lie algebra correspond bijectively to representations of its universal enveloping algebra. However, the representation theory of general associative algebras does not have all of the nice properties of the representation theory of groups and Lie algebras. Module theory When considering representations of an associative algebra, one can forget the underlying field, and simply regard the associative algebra as a ring, and its representations as modules. This approach is surprisingly fruitful: many results in representation theory can be interpreted as special cases of results about modules over a ring. Hopf algebras and quantum groups Hopf algebras provide a way to improve the representation theory of associative algebras, while retaining the representation theory of groups and Lie algebras as special cases. In particular, the tensor product of two representations is a representation, as is the dual vector space. The Hopf algebras associated to groups have a commutative algebra structure, and so general Hopf algebras are known as quantum groups, although this term is often restricted to certain Hopf algebras arising as deformations of groups or their universal enveloping algebras. The representation theory of quantum groups has added surprising insights to the representation theory of Lie groups and Lie algebras, for instance through the crystal basis of Kashiwara.

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Generalizations Set-theoretical representations A set-theoretic representation (also known as a group action or permutation representation) of a group G on a set X is given by a function ρ from G to XX, the set of functions from X to X, such that for all g1, g2 in G and all x in X:

This condition and the axioms for a group imply that ρ(g) is a bijection (or permutation) for all g in G. Thus we may equivalently define a permutation representation to be a group homomorphism from G to the symmetric group SX of X.

Representations in other categories Every group G can be viewed as a category with a single object; morphisms in this category are just the elements of G. Given an arbitrary category C, a representation of G in C is a functor from G to C. Such a functor selects an object X in C and a group homomorphism from G to Aut(X), the automorphism group of X. In the case where C is VectF, the category of vector spaces over a field F, this definition is equivalent to a linear representation. Likewise, a set-theoretic representation is just a representation of G in the category of sets. For another example consider the category of topological spaces, Top. Representations in Top are homomorphisms from G to the homeomorphism group of a topological space X. Two types of representations closely related to linear representations are: • projective representations: in the category of projective spaces. These can be described as "linear representations up to scalar transformations". • affine representations: in the category of affine spaces. For example, the Euclidean group acts affinely upon Euclidean space.

Representations of categories Since groups are categories, one can also consider representation of other categories. The simplest generalization is to monoids, which are categories with one object. Groups are monoids for which every morphism is invertible. General monoids have representations in any category. In the category of sets, these are monoid actions, but monoid representations on vector spaces and other objects can be studied. More generally, one can relax the assumption that the category being represented has only one object. In full generality, this is simply the theory of functors between categories, and little can be said. One special case has had a significant impact on representation theory, namely the representation theory of quivers.[11] A quiver is simply a directed graph (with loops and multiple arrows allowed), but it can be made into a category (and also an algebra) by considering paths in the graph. Representations of such categories/algebras have illuminated several aspects of representation theory, for instance by allowing non-semisimple representation theory questions about a group to be reduced in some cases to semisimple representation theory questions about a quiver.

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Notes [1] Classic texts on representation theory include Curtis & Reiner (1962) and Serre (1977). Other excellent sources are Fulton & Harris (1991) and Goodman & Wallach (1998). [2] For the history of the representation theory of finite groups, see Lam (1998). For algebraic and Lie groups, see Borel (2001). [3] There are many textbooks on vector spaces and linear algebra. For an advanced treatment, see Kostrikin & Manin (1997). [4] Sally & Vogan 1989. [5] Sternberg 1994. [6] Lam 1998, p. 372. [7] Folland 1995. [8] Goodman & Wallach 1998, Olver 1999, Sharpe 1997. [9] Borel & Casselman 1979, Gelbert 1984. [10] See the previous footnotes and also Borel (2001). [11] Simson, Skowronski & Assem 2007. [12] Fulton & Harris 1991, Simson, Skowronski & Assem 2007, Humphreys 1972. [13] This material can be found in standard textbooks, such as Curtis & Reiner (1962), Fulton & Harris (1991), Goodman & Wallach (1998), Gordon & Liebeck (1993), Humphreys (1972), Jantzen (2003), Knapp (2001) and Serre (1977). [14] The representation {0} of dimension zero is considered to be neither reducible nor irreducible, just like the number 1 is considered to be neither composite nor prime. [15] Alperin 1986, Lam 1998, Serre 1977. [16] Kim 1999. [17] Serre 1977, Part III [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36]

Alperin 1986. See Weyl 1928. Wigner 1939. Borel 2001. Knapp 2001. Peter & Weyl 1927. Bargmann 1947. Pontrjagin 1934. Weyl 1946. Fulton & Harris 1991. Humphreys 1972a. Kac 1990. Kac 1977. Humphreys 1972b, Jantzen 2003. Olver 1999. Mumford, Fogarty & Kirwan 1994. Sharpe 1997. Borel & Casselman 1979. Gelbart 1984.

References • Alperin, J. L. (1986), Local Representation Theory: Modular Representations as an Introduction to the Local Representation Theory of Finite Groups, Cambridge University Press, ISBN 978-0521449267. • Bargmann, V. (1947), "Irreducible unitary representations of the Lorenz group" (http://jstor.org/stable/ 1969129), Annals of Mathematics (Annals of Mathematics) 48 (3): 568–640, doi:10.2307/1969129. • Borel, Armand (2001), Essays in the History of Lie Groups and Algebraic Groups, American Mathematical Society, ISBN 978-0821802885. • Borel, Armand; Casselman, W. (1979), Automorphic Forms, Representations, and L-functions, American Mathematical Society, ISBN 978-0821814352. • Curtis, Charles W.; Reiner, Irving (1962), Representation Theory of Finite Groups and Associative Algebras, John Wiley & Sons (Reedition 2006 by AMS Bookstore), ISBN 978-0470189757 (ISBN 978-0821840665). • Gelbart, Stephen (1984), "An Elementary Introduction to the Langlands Program" (http://www.ams.org/bull/ 1984-10-02/S0273-0979-1984-15237-6/home.html), Bulletin of the American Mathematical Society 10 (2):

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• •

• • • • • •

177–219, doi:10.1090/S0273-0979-1984-15237-6. Folland, Gerald B. (1995), A Course in Abstract Harmonic Analysis, CRC Press, ISBN 978-0849384905. Fulton, William; Harris, Joe (1991), Representation theory. A first course, Graduate Texts in Mathematics, Readings in Mathematics, 129, New York: Springer-Verlag, MR1153249, ISBN 978-0-387-97527-6, ISBN 978-0-387-97495-8. Goodman, Roe; Wallach, Nolan R. (1998), Representations and Invariants of the Classical Groups, Cambridge University Press, ISBN 978-0521663489. Gordon, James; Liebeck, Martin (1993), Representations and Characters of Finite Groups, Cambridge: Cambridge University Press, ISBN 978-0-521-44590-0. Helgason, Sigurdur (1978), Differential Geometry, Lie groups and Symmetric Spaces, Academic Press, ISBN 978-0-12-338460-7 Humphreys, James E. (1972a), Introduction to Lie Algebras and Representation Theory, Birkhäuser, ISBN 978-0387900537. Humphreys, James E. (1972b), Linear Algebraic Groups, Graduate Texts in Mathematics, 21, Berlin, New York: Springer-Verlag, MR0396773, ISBN 978-0-387-90108-4 Jantzen, Jens Carsten (2003), Representations of Algebraic Groups, American Mathematical Society, ISBN 978-0821835272.

• Kac, Victor G. (1977), "Lie superalgebras", Advances in Mathematics 26 (1): 8–96, doi:10.1016/0001-8708(77)90017-2. • Kac, Victor G. (1990), Infinite Dimensional Lie Algebras (3rd ed.), Cambridge University Press, ISBN 978-0521466936. • Knapp, Anthony W. (2001), Representation Theory of Semisimple Groups: An Overview Based on Examples, Princeton University Press, ISBN 978-0691090894. • Kim, Shoon Kyung (1999), Group Theoretical Methods and Applications to Molecules and Crystals: And Applications to Molecules and Crystals, Cambridge University Press, ISBN 978-0521640626. • Kostrikin, A. I.; Manin, Yuri I. (1997), Linear Algebra and Geometry, Taylor & Francis, ISBN 978-9056990497. • Lam, T. Y. (1998), "Representations of finite groups: a hundred years", Notices of the AMS (American Mathematical Society) 45 (3,4): 361–372 (Part I) (http://www.ams.org/notices/199803/lam.pdf), 465–474 (Part II) (http://www.ams.org/notices/199804/lam2.pdf). • Yurii I. Lyubich. Introduction to the Theory of Banach Representations of Groups. Translated from the 1985 Russian-language edition (Kharkov, Ukraine). Birkhäuser Verlag. 1988. • Mumford, David; Fogarty, J.; Kirwan, F. (1994), Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], 34 (3rd ed.), Berlin, New York: Springer-Verlag, MR0214602(1st ed. 1965) MR0719371 (2nd ed.) MR1304906(3rd ed.), ISBN 978-3-540-56963-3 • Olver, Peter J. (1999), Classical invariant theory, Cambridge: Cambridge University Press, ISBN 0-521-55821-2. • Peter, F.; Weyl, Hermann (1927), "Die Vollständigkeit der primitiven Darstellungen einer geschlossenen kontinuierlichen Gruppe", Mathematische Annalen 97: 737–755, doi:10.1007/BF01447892. • Pontrjagin, Lev S. (1934), "The theory of topological commutative groups" (http://jstor.org/stable/1968438), Annals of Mathematics (Annals of Mathematics) 35 (2): 361–388, doi:10.2307/1968438. • Sally, Paul; Vogan, David A. (1989), Representation Theory and Harmonic Analysis on Semisimple Lie Groups, American Mathematical Society, ISBN 978-0821815267. • Serre, Jean-Pierre (1977), Linear Representations of Finite Groups, Springer-Verlag, ISBN 978-0387901909. • Sharpe, Richard W. (1997), Differential Geometry: Cartan's Generalization of Klein's Erlangen Program, Springer, ISBN 978-0387947327. • Simson, Daniel; Skowronski, Andrzej; Assem, Ibrahim (2007), Elements of the Representation Theory of Associative Algebras, Cambridge University Press, ISBN 978-0521882187.

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Symmetry in physics In physics, symmetry includes all features of a physical system that exhibit the property of symmetry—that is, under certain transformations, aspects of these systems are "unchanged", according to a particular observation. A symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is "preserved" under some change. The transformations may be continuous (such as rotation of a circle) or discrete (e.g., reflection of a bilaterally symmetric figure, or rotation of a regular polygon). Continuous and discrete transformations give rise to corresponding types of symmetries. Continuous symmetries can be described by Lie groups while discrete symmetries are described by finite groups (see Symmetry group). Symmetries are frequently amenable to mathematical formulation and can be exploited to simplify many problems. An important example of such symmetry is the invariance of the form of physical laws under arbitrary differentiable coordinate transformations.

Symmetry as invariance Invariance is specified mathematically by transformations that leave some quantity unchanged. This idea can apply to basic real-world observations. For example, temperature may be constant throughout a room. Since the temperature is independent of position within the room, the temperature is invariant under a shift in the measurer's position. Similarly, a uniform sphere rotated about its center will appear exactly as it did before the rotation. The sphere is said to exhibit spherical symmetry. A rotation about any axis of the sphere will preserve how the sphere "looks".

Invariance in force The above ideas lead to the useful idea of invariance when discussing observed physical symmetry; this can be applied to symmetries in forces as well. For example, an electrical wire is said to exhibit cylindrical symmetry, because the electric field strength at a given distance r from an electrically charged wire of infinite length will have the same magnitude at each point on the surface of a cylinder (whose axis is the wire) with radius r. Rotating the wire about its own axis does not change its position, hence it will preserve the field. The field strength at a rotated position is the same, but its direction is rotated accordingly. These two properties are interconnected through the more general property that rotating any system of charges causes a corresponding rotation of the electric field. In Newton's theory of mechanics, given two bodies, each with mass m, starting from rest at the origin and moving along the x-axis in opposite directions, one with speed v1 and the other with speed v2 the total kinetic energy of the system (as calculated from an observer at the origin) is 1⁄2m(v12 + v22) and remains the same if the velocities are interchanged. The total kinetic energy is preserved under a reflection in the y-axis.

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The last example above illustrates another way of expressing symmetries, namely through the equations that describe some aspect of the physical system. The above example shows that the total kinetic energy will be the same if v1 and v2 are interchanged.

Local and global symmetries Symmetries may be broadly classified as global or local. A global symmetry is one that holds at all points of spacetime, whereas a local symmetry is one that has a different symmetry transformation at different points of spacetime; specifically a local symmetry transformation is parameterised by the spacetime co-ordinates. Local symmetries play an important role in physics as they form the basis for gauge theories.

Continuous symmetries The two examples of rotational symmetry described above - spherical and cylindrical - are each instances of continuous symmetry. These are characterised by invariance following a continuous change in the geometry of the system. For example, the wire may be rotated through any angle about its axis and the field strength will be the same on a given cylinder. Mathematically, continuous symmetries are described by continuous or smooth functions. An important subclass of continuous symmetries in physics are spacetime symmetries.

Spacetime symmetries Continuous spacetime symmetries are symmetries involving transformations of space and time. These may be further classified as spatial symmetries, involving only the spatial geometry associated with a physical system; temporal symmetries, involving only changes in time; or spatio-temporal symmetries, involving changes in both space and time. • Time translation: A physical system may have the same features over a certain interval of time expressed mathematically as invariance under the transformation

; this is

for any real numbers t and a in the

interval. For example, in classical mechanics, a particle solely acted upon by gravity will have gravitational potential energy when suspended from a height above the Earth's surface. Assuming no change in the height of the particle, this will be the total gravitational potential energy of the particle at all times. In other words, by considering the state of the particle at some time (in seconds) and also at , say, the particle's total gravitational potential energy will be preserved. • Spatial translation: These spatial symmetries are represented by transformations of the form

and

describe those situations where a property of the system does not change with a continuous change in location. For example, the temperature in a room may be independent of where the thermometer is located in the room. • Spatial rotation: These spatial symmetries are classified as proper rotations and improper rotations. The former are just the 'ordinary' rotations; mathematically, they are represented by square matrices with unit determinant. The latter are represented by square matrices with determinant -1 and consist of a proper rotation combined with a spatial reflection (inversion). For example, a sphere has proper rotational symmetry. Other types of spatial rotations are described in the article Rotation symmetry. • Poincaré transformations: These are spatio-temporal symmetries which preserve distances in Minkowski spacetime, i.e. they are isometries of Minkowski space. They are studied primarily in special relativity. Those isometries that leave the origin fixed are called Lorentz transformations and give rise to the symmetry known as Lorentz covariance. • Projective symmetries: These are spatio-temporal symmetries which preserve the geodesic structure of spacetime. They may be defined on any smooth manifold, but find many applications in the study of exact solutions in general relativity.

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• Inversion transformations: These are spatio-temporal symmetries which generalise Poincaré transformations to include other conformal one-to-one transformations on the space-time coordinates. Lengths are not invariant under inversion transformations but there is a cross-ratio on four points that is invariant. Mathematically, spacetime symmetries are usually described by smooth vector fields on a smooth manifold. The underlying local diffeomorphisms associated with the vector fields correspond more directly to the physical symmetries, but the vector fields themselves are more often used when classifying the symmetries of the physical system. Some of the most important vector fields are Killing vector fields which are those spacetime symmetries that preserve the underlying metric structure of a manifold. In rough terms, Killing vector fields preserve the distance between any two points of the manifold and often go by the name of isometries. The article Isometries in physics discusses these symmetries in more detail.

Discrete symmetries A discrete symmetry is a symmetry that describes non-continuous changes in a system. For example, a square possesses discrete rotational symmetry, as only rotations by multiples of right angles will preserve the square's original appearance. Discrete symmetries sometimes involve some type of 'swapping', these swaps usually being called reflections or interchanges. • Time reversal: Many laws of physics describe real phenomena when the direction of time is reversed. Mathematically, this is represented by the transformation, . For example, Newton's second law of motion still holds if, in the equation

,

is replaced by

. This may be illustrated by describing the

motion of a particle thrown up vertically (neglecting air resistance). For such a particle, position is symmetric with respect to the instant that the object is at its maximum height. Velocity at reversed time is reversed. • Spatial inversion: These are represented by transformations of the form

and indicate an invariance

property of a system when the coordinates are 'inverted'. • Glide reflection: These are represented by a composition of a translation and a reflection. These symmetries occur in some crystals and in some planar symmetries, known as wallpaper symmetries.

C, P, and T symmetries The Standard model of particle physics has three related natural near-symmetries. These state that the actual universe about us is indistinguishable from one where: • Every particle is replaced with its antiparticle. This is C-symmetry (charge symmetry); • Everything appears as if reflected in a mirror. This is P-symmetry (parity symmetry); • The direction of time is reversed. This is T-symmetry (time symmetry). T-symmetry is counterintuitive (surely the future and the past are not symmetrical) but explained by the fact that the Standard model describes local properties, not global ones like entropy. To properly reverse the direction of time, one would have to put the big bang and the resulting low-entropy state in the "future." Since we perceive the "past" ("future") as having lower (higher) entropy than the present (see perception of time), the inhabitants of this hypothetical time-reversed universe would perceive the future in the same way as we perceive the past. These symmetries are near-symmetries because each is broken in the present-day universe. However, the Standard Model predicts that the combination of the three (that is, the simultaneous application of all three transformations) must be a symmetry, called CPT symmetry. CP violation, the violation of the combination of C- and P-symmetry, is necessary for the presence of significant amounts of baryonic matter in the universe and thus is a prerequisite for the existence of life. CP violation is a fruitful area of current research in particle physics.

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Supersymmetry A type of symmetry known as supersymmetry has been used to try to make theoretical advances in the standard model. Supersymmetry is based on the idea that there is another physical symmetry beyond those already developed in the standard model, specifically a symmetry between bosons and fermions. Supersymmetry asserts that each type of boson has, as a supersymmetric partner, a fermion, called a superpartner, and vice versa. Supersymmetry has not yet been experimentally verified: no known particle has the correct properties to be a superpartner of any other known particle. If superpartners exist they must have masses greater than current particle accelerators can generate.

Mathematics of physical symmetry The transformations describing physical symmetries typically form a mathematical group. Group theory is an important area of mathematics for physicists. Continuous symmetries are specified mathematically by continuous groups (called Lie groups). Many physical symmetries are isometries and are specified by symmetry groups. Sometimes this term is used for more general types of symmetries. The set of all proper rotations (about any angle) through any axis of a sphere form a Lie group called the special orthogonal group . (The 3 refers to the three-dimensional space of an ordinary sphere.) Thus, the symmetry group of the sphere with proper rotations is

. Any rotation preserves distances on the surface of

the ball. The set of all Lorentz transformations form a group called the Lorentz group (this may be generalised to the Poincaré group). Discrete symmetries are described by discrete groups. For example, the symmetries of an equilateral triangle are described by the symmetric group . An important type of physical theory based on local symmetries is called a gauge theory and the symmetries natural to such a theory are called gauge symmetries. Gauge symmetries in the Standard model, used to describe three of the fundamental interactions, are based on the SU(3) × SU(2) × U(1) group. (Roughly speaking, the symmetries of the SU(3) group describe the strong force, the SU(2) group describes the weak interaction and the U(1) group describes the electromagnetic force.) Also, the reduction by symmetry of the energy functional under the action by a group and spontaneous symmetry breaking of transformations of symmetric groups appear to elucidate topics in particle physics (for example, the unification of electromagnetism and the weak force in physical cosmology).

Conservation laws and symmetry The symmetry properties of a physical system are intimately related to the conservation laws characterizing that system. Noether's theorem gives a precise description of this relation. The theorem states that each continuous symmetry of a physical system implies that some physical property of that system is conserved. Conversely, each conserved quantity has a corresponding symmetry. For example, the isometry of space gives rise to conservation of (linear) momentum, and isometry of time gives rise to conservation of energy. The following table summarizes some fundamental symmetries and the associated conserved quantity.

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Class

Invariance

Conserved quantity

Proper orthochronous Lorentz symmetry

translation in time (homogeneity)

energy

translation in space (homogeneity)

linear momentum

rotation in space (isotropy)

angular momentum

P, coordinate inversion

spatial parity

C, charge conjugation

charge parity

T, time reversal

time parity

CPT

product of parities

U(1) gauge transformation

electric charge

U(1) gauge transformation

lepton generation number

U(1) gauge transformation

hypercharge

U(1)Y gauge transformation

weak hypercharge

U(2) [U(1)xSU(2)]

electroweak force

SU(2) gauge transformation

isospin

Discrete symmetry

Internal symmetry (independent of spacetime coordinates)

SU(2)L gauge transformation weak isospin PxSU(2)

G-parity

SU(3) "winding number"

baryon number

SU(3) gauge transformation

quark color

SU(3) (approximate)

quark flavor

S((U2)xU(3)) [ U(1)xSU(2)xSU(3)]

Standard Model

References General readers • Leon Lederman and Christopher T. Hill (2005) Symmetry and the Beautiful Universe. Amherst NY: Prometheus Books. • Schumm, Bruce (2004) Deep Down Things. Johns Hopkins Univ. Press. • Victor J. Stenger (2000) Timeless Reality: Symmetry, Simplicity, and Multiple Universes. Buffalo NY: Prometheus Books. Chpt. 12 is a gentle introduction to symmetry, invariance, and conservation laws. • Anthony Zee (2007) Fearful Symmetry: The search for beauty in modern physics, [1] 2nd ed. Princeton University Press. ISBN 978-0-691-00946-9. 1986 1st ed. published by Macmillan.

Symmetry in physics

Technical • Brading, K., and Castellani, E., eds. (2003) Symmetries in Physics: Philosophical Reflections. Cambridge Univ. Press. • -------- (2007) "Symmetries and Invariances in Classical Physics" in Butterfield, J., and John Earman, eds., Philosophy of Physic Part B. North Holland: 1331-68. • Debs, T. and Redhead, M. (2007) Objectivity, Invariance, and Convention: Symmetry in Physical Science. Harvard Univ. Press. • John Earman (2002) "Laws, Symmetry, and Symmetry Breaking: Invariance, Conservations Principles, and Objectivity. [2]" Address to the 2002 meeting of the Philosophy of Science Association. • Mainzer, K. (1996) Symmetries of nature. Berlin: De Gruyter. • Thompson, William J. (1994) Angular Momentum: An Illustrated Guide to Rotational Symmetries for Physical Systems. Wiley. ISBN 0-471-55264. • Bas Van Fraassen (1989) Laws and symmetry. Oxford Univ. Press. • Eugene Wigner (1967) Symmetries and Reflections. Indiana Univ. Press.

External links • Stanford Encyclopedia of Philosophy: "Symmetry [3]" -- by K. Brading and E. Castellani.

References [1] http:/ / press. princeton. edu/ titles/ 8509. html [2] http:/ / philsci-archive. pitt. edu/ archive/ 00000878/ 00/ PSA2002. pdf [3] http:/ / plato. stanford. edu/ entries/ symmetry-breaking/

Space group In crystallography, the space group (or crystallographic group, or Fedorov group) of a crystal is a description of the symmetry of the crystal, and can have one of 230 types. In mathematics space groups are also studied in dimensions other than 3 where they are sometimes called Bieberbach groups, and are discrete cocompact groups of isometries of an oriented Euclidean space. A definitive source regarding 3-dimensional space groups is the International Tables for Crystallography (Hahn (2002)).

History The space groups in 3 dimensions were first enumerated by Fyodorov (1891), and shortly afterwards were independently enumerated by Barlow (1894) and Schönflies (1891). These first enumerations all contained several minor mistakes, and the correct list of 230 space groups was found during correspondence between Fyodorov and Schönflies. Space groups in 2 dimensions are the 17 wallpaper groups which have been known for several centuries.

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Elements of a space group The space groups in three dimensions are made from combinations of the 32 crystallographic point groups with the 14 Bravais lattices which belong to one of 7 lattice systems. This results in a space group being some combination of the translational symmetry of a unit cell including lattice centering, the point group symmetry operations of reflection, rotation and improper rotation (also called rotoinversion), and the screw axis and glide plane symmetry operations. The combination of all these symmetry operations results in a total of 230 unique space groups describing all possible crystal symmetries.

Elements fixing a point The elements of the space group fixing a point of space are rotations, reflections, the identity element, and improper rotations.

Translations The translations form a normal abelian subgroup of rank 3, called the Bravais lattice. There are 14 possible types of Bravais lattice. The quotient of the space group by the Bravais lattice is a finite group which is one of the 32 possible point groups.

Glide planes A glide plane is a reflection in a plane, followed by a translation parallel with that plane. This is noted by a, b or c, depending on which axis the glide is along. There is also the n glide, which is a glide along the half of a diagonal of a face, and the d glide, which is a fourth of the way along either a face or space diagonal of the unit cell. The latter is called the diamond glide plane as it features in the diamond structure.

Screw axes A screw axis is a rotation about an axis, followed by a translation along the direction of the axis. These are noted by a number, n, to describe the degree of rotation, where the number is how many operations must be applied to complete a full rotation (e.g., 3 would mean a rotation one third of the way around the axis each time). The degree of translation is then added as a subscript showing how far along the axis the translation is, as a portion of the parallel lattice vector. So, 21 is a twofold rotation followed by a translation of 1/2 of the lattice vector.

Notation for space groups There are at least eight methods of naming space groups. Some of these methods can assign several different names to the same space group, so altogether there are many thousands of different names. • Number. The International Union of Crystallography publishes tables of all space group types, and assigns each a unique number from 1 to 230. The numbering is arbitrary, except that groups with the same crystal system or point group are given consecutive numbers. • International symbol or Hermann-Mauguin notation. The Hermann-Mauguin (or international) notation describes the lattice and some generators for the group. It has a shortened form called the international short symbol, which is the one most commonly used in crystallography, and usually consists of a set of four symbols. The first describes the centering of the Bravais lattice (P, A, B, C, I, R or F). The next three describe the most prominent symmetry operation visible when projected along one of the high symmetry directions of the crystal. These symbols are the same as used in point groups, with the addition of glide planes and screw axis, described above. By way of example, the space group of quartz is P3121, showing that it exhibits primitive centering of the motif (i.e., once per unit cell), with a threefold screw axis and a twofold rotation axis. Note that it does not explicitly contain the crystal system, although this is unique to each space group (in the case of P3121, it is

187

Space group

188

trigonal). In the international short symbol the first symbol (31 in this example) denotes the symmetry along the major axis (c-axis in trigonal cases), the second (2 in this case) along axes of secondary importance (a and b) and the third symbol the symmetry in another direction. In the trigonal case there also exists a space group P3112. In this space group the twofold axes are not along the a and b-axes but in a direction rotated by 30o. The international symbols and international short symbols for some of the space groups were changed slightly between 1935 and 2002, so several space groups have 4 different international symbols in use. • Hall notation. Space group notation with an explicit origin. Rotation, translation and axis-direction symbols are clearly separated and inversion centers are explicitly defined. The construction and format of the notation make it particularly suited to computer generation of symmetry information. For example, group number 3 has three Hall symbols: P 2y (P 1 2 1), P 2 (P 1 1 2), P 2x (P 2 1 1). • Schönflies notation. The space groups with given point group are numbered by 1, 2, 3, ... (in the same order as their international number) and this number is added as a superscript to the Schönflies symbol for the point group. For example, groups numbers 3 to 5 whose point group is C2 have Schönflies symbols C12, C22, C32. • Shubnikov symbol • 2D:Orbifold notation and 3D:Fibrifold notation. As the name suggests, the orbifold notation describes the orbifold, given by the quotient of Euclidean space by the space group, rather than generators of the space group. It was introduced by Conway and Thurston, and is not used much outside mathematics. Some of the space groups have several different fibrifolds associated to them, so have several different fibrifold symbols.

Classification systems for space groups There are (at least) 10 different ways to classify space groups into classes. The relations between some of these are described in the following table. Each classification system is a refinement of the ones below it. (Crystallographic) space group types (230 in three dimensions). Two space groups, considered as subgroups of the group of affine transformations of space, have the same space group type if they are conjugate by an orientation-preserving affine transformation. In three dimensions,for 11 of the affine space groups, there is no orientation-preserving map from the group to its mirror image, so if one distinguishes groups from their mirror images these each split into two cases. So there are 54+11=65 space group types that preserve orientation. Affine space group types (219 in three dimensions). Two space groups, considered as subgroups of the group of affine transformations of space, have the same affine space group type if they are conjugate under an affine transformation. The affine space group type is determined by the underlying abstract group of the space group. In three dimensions there are 54 affine space group types that preserve orientation. Arithmetic crystal classes (73 in three dimensions). These are determined by the point group together with the action of the point group on the subgroup of translations. In other words the arithmetic crystal classes correspond to conjugacy classes of finite subgroup of the general linear group GLn(Z) over the integers. A space group is called symmorphic (or split) if there is a point such that all symmetries are the product of asymmetry fixing this point and a translation. Equivalently, a space group is symmorphic if it is a semidirect product of its point group with its translation subgroup. There are 73 symmorphic space groups, with exactly one in each arithmetic crystal class. There are also 157 nonsymmorphic space group types with varying numbers in the arithmetic crystal classes. (geometric) Crystal classes (32 in three dimensions). The crystal class of a space group is determined by its point group: the quotient by the subgroup of translations, acting on the lattice. Two space groups are in the same crystal class if and only if their point groups, which are subgroups of GL2(Z), are conjugate in the larger group GL2(Q).

Bravais flocks (14 in three dimensions). These are determined by the underlying Bravais lattice type. These correspond to conjugacy classes of lattice point groups in GL2(Z), where the lattice point group is the group of symmetries of the underlying lattice that fix a point of the lattice, and contains the point group.

Space group

Crystal systems. (7 in three dimensions) Crystal systems are an ad hoc modification of the lattice systems to make them compatible with the classification according to point groups. They differ from crystal families in that the hexagonal crystal family is split into two subsets, called the trigonal and hexagonal crystal systems. The trigonal crystal system is larger than the rhombohedral lattice system, the hexagonal crystal system is smaller than the hexagonal lattice system, and the remaining crystal systems and lattice systems are the same.

189 Lattice systems (7 in three dimensions). The lattice system of a space group is determined by the conjugacy class of the lattice point group (a subgroup of GL2(Z)) in the larger group GL2(Q). In three dimensions the lattice point group can have one of the 7 different orders 2, 4, 8, 12, 16, 24, or 48. The hexagonal crystal family is split into two subsets, called the rhombohedral and hexagonal lattice systems.

Crystal families (6 in three dimensions). The point group of a space group does not quite determine its lattice system, because occasionally two space groups with the same point group may be in different lattice systems. Crystal families are formed from lattice systems by merging the two lattice systems whenever this happens, so that the crystal family of a space group is determined by either its lattice system or its point group. In 3 dimensions the only two lattice families that get merged in this way are the hexagonal and rhombohedral lattice systems, which are combined into the hexagonal crystal family. The 6 crystal families in 3 dimensions are called triclinic, monoclinic, orthorhombal, tetragonal, hexagonal, and cubic. Crystal families are commonly used in popular books on crystals, where they are sometimes called crystal systems.

Conway, Delgado Friedrichs, and Huson et al. (2001) gave another classification of the space groups, according to the fibrifold structures on the corresponding orbifold. They divided the 219 affine space groups into reducible and irreducible groups. The reducible groups fall into 17 classes corresponding to the 17 wallpaper groups, and the remaining 35 irreducible groups are the same as the cubic groups and are classified separately.

Space groups in other dimensions Bieberbach's theorems In n dimensions, an affine space group, or Bieberbach group, is a discrete subgroup of isometries of n-dimensional Euclidean space with a compact fundamental domain. Bieberbach (1911, 1912) proved that the subgroup of translations of any such group contains n linearly independent translations, and is a free abelian subgroup of finite index, and is also the unique maximal normal abelian subgroup. He also showed that in any dimension n there are only a finite number of possibilities for the isomorphism class of the underlying group of a space group, and moreover the action of the group on Euclidean space is unique up to conjugation by affine transformations. This answers part of Hilbert's 18th problem. Zassenhaus (1948) showed that conversely any group that is the extension of Zn by a finite group acting faithfully is an affine space group. Combining these results shows that classifying space groups in n dimensions up to conjugation by affine transformations is essentially the same as classifying isomorphism classes for groups that are extensions of Zn by a finite group acting faithfully. It is essential in Bieberbach's theorems to assume that the group acts as isometries; the theorems do not generalize to discrete cocompact groups of affine transformations of Euclidean space. A counter-example is given by the 3-dimensional Heisenberg group of the integers acting by translations on the Heisenberg group of the reals, identified with 3-dimensional Euclidean space. This is a discrete cocompact group of affine transformations of space, but does not contain a subgroup Z3.

Classification in small dimensions This table give the number of space group types in small dimensions.

Space group

190

Dimension Lattice types (sequence [1] A004030 in OEIS)

point groups (sequence [2] A004028 in OEIS)

Classification

Crystallographic Affine space space group types group types [3] (sequence A006227 (sequence [4] in OEIS) A004029 in OEIS)

0

1

1

1

1

Trivial group

1

1

2

2

2

One is the group of integers and the other is the infinite dihedral group;see symmetry groups in one dimension

2

5

10

17

17

these 2D space groups are also called wallpaper groups or plane groups.

3

14

32

230

219

In 3D there are 230 crystallographic space group types, which reduces to 219 affine space group types because of some types being different from their mirror image; these are said to differ by "enantiomorphous character" (e.g. P3112 and P3212). Usually "space group" refers to 3D. They were enumerated independently by Barlow (1894), Fedorov (1891) and Schönflies (1891).

4

64

227

4895

4783

The 4895 4-dimensional groups were enumerated by Harold Brown, Rolf Bülow, and Joachim Neubüser et al. (1978).

5

189

955

222018

Plesken & Schulz (2000) enumerated the ones of dimension 5

28927922

Plesken & Schulz (2000) enumerated the ones of dimension 6

6

7104

28934974

Double groups and time reversal In addition to crystallographic space groups there are also magnetic space groups or double groups. These symmetries contain an element known as time reversal. They are of importance in magnetic structures that contain ordered unpaired spins, i.e. ferro-, ferri- or antiferromagnetic structures as studied by neutron diffraction. The time reversal element flips a magnetic spin while leaving all other structure the same and it can be combined with a number of other symmetry elements. Including time reversal there are 1651 magnetic space groups in 3D (Kim 1999, p.428).

Table of space groups in 3 dimensions Crystal system

Point group

#

Space groups (international short symbol)

Hermann-Mauguin Schönflies Triclinic (2)

Monoclinic (13)

1

C1

1

P1

1

Ci

2

P1

2

C2

3-5

P2, P21, C2

m

Cs

6-9

Pm, Pc, Cm, Cc

2/m

C2h

10-15

P2/m, P21/m, C2/m, P2/c, P21/c, C2/c

Space group

Orthorhombic (59)

Tetragonal (68)

Trigonal (25)

Hexagonal (27)

Cubic (36)

191 222

D2

16-24

P222, P2221, P21212, P212121, C2221, C222, F222, I222, I212121

mm2

C2v

25-46

Pmm2, Pmc21, Pcc2, Pma2, Pca21, Pnc2, Pmn21, Pba2, Pna21, Pnn2, Cmm2, Cmc21, Ccc2, Amm2, Aem2, Ama2, Aea2, Fmm2, Fdd2, Imm2, Iba2, Ima2

mmm

D2h

47-74

Pmmm, Pnnn, Pccm, Pban, Pmma, Pnna, Pmna, Pcca, Pbam, Pccn, Pbcm, Pnnm, Pmmn, Pbcn, Pbca, Pnma, Cmcm, Cmce, Cmmm, Cccm, Cmme, Ccce, Fmmm, Fddd, Immm, Ibam, Ibca, Imma

4

C4

75-80

P4, P41, P42, P43, I4, I41

4

S4

81-82

P4, I4

4/m

C4h

83-88

P4/m, P42/m, P4/n, P42/n, I4/m, I41/a

422

D4

89-98

P422, P4212, P4122, P41212, P4222, P42212, P4322, P43212, I422, I4122

4mm

C4v

99-110

P4mm, P4bm, P42cm, P42nm, P4cc, P4nc, P42mc, P42bc, I4mm, I4cm, I41md, I41cd

42m

D2d

111-122

P42m, P42c, P421m, P421c, P4m2, P4c2, P4b2, P4n2, I4m2, I4c2, I42m, I42d

4/mmm

D4h

123-142

P4/mmm, P4/mcc, P4/nbm, P4/nnc, P4/mbm, P4/mnc, P4/nmm, P4/ncc, P42/mmc, P42/mcm, P42/nbc, P42/nnm, P42/mbc, P42/mnm, P42/nmc, P42/ncm, I4/mmm, I4/mcm, I41/amd, I41/acd

3

C3

143-146

P3, P31, P32, R3

3

S6

147-148

P3, R3

32

D3

149-155

P312, P321, P3112, P3121, P3212, P3221, R32

3m

C3v

156-161

P3m1, P31m, P3c1, P31c, R3m, R3c

3m

D3d

162-167

P31m, P31c, P3m1, P3c1, R3m, R3c,

6

C6

168-173

P6, P61, P65, P62, P64, P63

6

C3h

174

P6

6/m

C6h

175-176

P6/m, P63/m

622

D6

177-182

P622, P6122, P6522, P6222, P6422, P6322

6mm

C6v

183-186

P6mm, P6cc, P63cm, P63mc

6m2

D3h

187-190

P6m2, P6c2, P62m, P62c

6/mmm

D6h

191-194

P6/mmm, P6/mcc, P63/mcm, P63/mmc

23

T

195-199

P23, F23, I23, P213, I213

m3

Th

200-206

Pm3, Pn3, Fm3, Fd3, Im3, Pa3, Ia3

432

O

207-214

P432, P4232, F432, F4132, I432, P4332, P4132, I4132

43m

Td

215-220

P43m, F43m, I43m, P43n, F43c, I43d

m3m

Oh

221-230

Pm3m, Pn3n, Pm3n, Pn3m, Fm3m, Fm3c, Fd3m, Fd3c, Im3m, Ia3d

Note. An e plane is a double glide plane, one having glides in two different directions. They are found in five space groups, all in the orthorhombic system and with a centered lattice. The use of the symbol e became official with Hahn (2002). The lattice system can be found as follows. If the crystal system is not trigonal then the lattice system is of the same type. If the crystal system is trigonal, then the lattice system is hexagonal unless the space group is one of the seven in the rhombohedral lattice system consisting of the 7 trigonal space groups in the table above whose name begins with R. (The term rhombohedral system is also sometimes used as an alternative name for the whole trigonal system.) The hexagonal lattice system is larger than the hexagonal crystal system, and consists of the hexagonal crystal system together with the 18 groups of the trigonal crystal system other than the seven whose names begin with R.

Space group The Bravais lattice of the space group is determined by the lattice system together with the initial letter of its name, which for the non-rhombohedral groups is P, I, F, or C, standing for the principal, body centered, face centered, or C-face centered lattices.

References • Barlow, W (1894), "Über die geometrischen Eigenschaften starrer Strukturen und ihre Anwendung auf Kristalle", Z. Kristallogr. 23: 1–63 • Bieberbach, Ludwig (1911), "Über die Bewegungsgruppen der Euklidischen Räume", Mathematische Annalen 70 (3): 297–336, doi:10.1007/BF01564500, ISSN 0025-5831 • Bieberbach, Ludwig (1912), "Über die Bewegungsgruppen der Euklidischen Räume (Zweite Abhandlung.) Die Gruppen mit einem endlichen Fundamentalbereich", Mathematische Annalen 72 (3): 400–412, doi:10.1007/BF01456724, ISSN 0025-5831 • Brown, Harold; Bülow, Rolf; Neubüser, Joachim; Wondratschek, Hans; Zassenhaus, Hans (1978), Crystallographic groups of four-dimensional space, New York: Wiley-Interscience [John Wiley & Sons], MR0484179, ISBN 978-0-471-03095-9 • Burckhardt, Johann Jakob (1947), Die Bewegungsgruppen der Kristallographie, Lehrbücher und Monographien aus dem Gebiete der exakten Wissenschaften, 13, Verlag Birkhäuser, Basel, MR0020553 • Conway, John Horton; Delgado Friedrichs, Olaf; Huson, Daniel H.; Thurston, William P. (2001), "On three-dimensional space groups" [5], Beiträge zur Algebra und Geometrie. Contributions to Algebra and Geometry 42 (2): 475–507, MR1865535, ISSN 0138-4821 • Fedorov, E. S. (1891), "Symmetry of Regular Systems of Figures", Zap. Mineral. Obch. 28 (2): 1–146 • Fedorov, E. S. (1971), Symmetry of crystals, ACA Monograph, 7, American Crystallographic Association • Hahn, Th. (2002), Hahn, Theo, ed., International Tables for Crystallography, Volume A: Space Group Symmetry [6] , A (5th ed.), Berlin, New York: Springer-Verlag, doi:10.1107/97809553602060000100, ISBN 978-0-7923-6590-7 • Hall, S.R. (1981), "Space-Group Notation with an Explicit Origin", Acta Cryst. A37: 517–525 • Kim, Shoon K. (1999), Group theoretical methods and applications to molecules and crystals, Cambridge University Press, MR1713786, ISBN 978-0-521-64062-6 • Plesken, Wilhelm; Schulz, Tilman (2000), "Counting crystallographic groups in low dimensions" [7], Experimental Mathematics 9 (3): 407–411, MR1795312, ISSN 1058-6458 • Schönflies, Arthur Moritz (1891), "Theorie der Kristallstruktur", Gebr. Bornträger, Berlin. • Vinberg, E. (2001), "Crystallographic group" [8], in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104 • Zassenhaus, Hans (1948), "Über einen Algorithmus zur Bestimmung der Raumgruppen" [9], Commentarii Mathematici Helvetici 21: 117–141, doi:10.1007/BF02568029, MR0024424, ISSN 0010-2571

External links • • • • • • •

International Union of Crystallography [10] Point Groups and Bravais Lattices [11] Bilbao Crystallographic Server [12] Space Group Info (old) [13] Space Group Info (new) [14] Crystal Lattice Structures: Index by Space Group [15] Full list of 230 crystallographic space groups [16]

• Interactive 3D visualization of all 230 crystallographic space groups [17] • Huson, Daniel H. (1999), The Fibrifold Notation and Classification for 3D Space Groups [18]

192

Space group

193

References [1] http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa004030 [2] http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa004028 [3] http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa006227 [4] http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa004029 [5] http:/ / www. emis. de/ journals/ BAG/ vol. 42/ no. 2/ 17. html [6] http:/ / it. iucr. org/ A/ [7] http:/ / projecteuclid. org/ euclid. em/ 1045604675 [8] http:/ / eom. springer. de/ C/ c027190. htm [9] http:/ / www. digizeitschriften. de/ index. php?id=166& ID=380406 [10] http:/ / www. iucr. org [11] http:/ / neon. mems. cmu. edu/ degraef/ pointgroups/ [12] http:/ / www. cryst. ehu. es/ [13] http:/ / cci. lbl. gov/ sginfo/ [14] http:/ / cci. lbl. gov/ cctbx/ explore_symmetry. html [15] http:/ / cst-www. nrl. navy. mil/ lattice/ spcgrp/ [16] http:/ / img. chem. ucl. ac. uk/ sgp/ mainmenu. htm [17] http:/ / www. spacegroup. info/ html/ space_groups. html [18] http:/ / www-ab. informatik. uni-tuebingen. de/ talks/ pdfs/ Fibrifolds-Princeton%201999. pdf

Molecular symmetry Molecular symmetry in chemistry describes the symmetry present in molecules and the classification of molecules according to their symmetry. Molecular symmetry is a fundamental concept in chemistry, as it can predict or explain many of a molecule's chemical properties, such as its dipole moment and its allowed spectroscopic transitions (based on selection rules such as the Laporte rule). Virtually every university level textbook on physical chemistry, quantum chemistry, and inorganic chemistry devotes a chapter to symmetry.[1] [2] [3] [4] [5] While various frameworks for the study of molecular symmetry exist, group theory is the predominant one. This framework is also useful in studying the symmetry of molecular orbitals, with applications such as the Hückel method, ligand field theory, and the Woodward-Hoffmann rules. Another framework on a larger scale is the use of crystal systems to describe crystallographic symmetry in bulk materials. Many techniques exist for the practical assessment of molecular symmetry, including X-ray crystallography and various forms of spectroscopy. Spectroscopic notation is based on symmetry considerations.

Symmetry concepts The study of symmetry in molecules is an adaptation of mathematical group theory.

Elements The symmetry of a molecule can be described by 5 types of symmetry elements. • Symmetry axis: an axis around which a rotation by

results in a molecule indistinguishable from the original.

This is also called an n-fold rotational axis and abbreviated Cn. Examples are the C2 in water and the C3 in ammonia. A molecule can have more than one symmetry axis; the one with the highest n is called the principal axis, and by convention is assigned the z-axis in a Cartesian coordinate system. • Plane of symmetry: a plane of reflection through which an identical copy of the original molecule is given. This is also called a mirror plane and abbreviated σ. Water has two of them: one in the plane of the molecule itself and one perpendicular to it. A symmetry plane parallel with the principal axis is dubbed vertical (σv) and one perpendicular to it horizontal (σh). A third type of symmetry plane exists: if a vertical symmetry plane additionally bisects the angle between two 2-fold rotation axes perpendicular to the principal axis, the plane is

Molecular symmetry

194

dubbed dihedral (σd). A symmetry plane can also be identified by its Cartesian orientation, e.g., (xz) or (yz). • Center of symmetry or inversion center, abbreviated i. A molecule has a center of symmetry when, for any atom in the molecule, an identical atom exists diametrically opposite this center an equal distance from it. There may or may not be an atom at the center. Examples are xenon tetrafluoride (XeF4) where the inversion center is at the Xe atom, and benzene (C6H6) where the inversion center is at the center of the ring. • Rotation-reflection axis: an axis around which a rotation by

, followed by a reflection in a plane

perpendicular to it, leaves the molecule unchanged. Also called an n-fold improper rotation axis, it is abbreviated Sn, with n necessarily even. Examples are present in tetrahedral silicon tetrafluoride, with three S4 axes, and the staggered conformation of ethane with one S6 axis. • Identity, abbreviated to E, from the German 'Einheit' meaning Unity.[6] This symmetry element simply consists of no change: every molecule has this element. While this element seems physically trivial, its consideration is necessary for the group-theoretical machinery to work properly. It is so called because it is analogous to multiplying by one (unity).

Operations The 5 symmetry elements have associated with them 5 symmetry operations. They are often, although not always, distinguished from the respective elements by a caret. Thus Ĉn is the rotation of a molecule around an axis and Ê is the identity operation. A symmetry element can have more than one symmetry operation associated with it. Since C1 is equivalent to E, S1 to σ and S2 to i, all symmetry operations can be classified as either proper or improper rotations.

Point groups A point group is a set of symmetry operations forming a mathematical group, for which at least one point remains fixed under all operations of the group. A crystallographic point group is a point group which is compatible with translational symmetry in three dimensions. There are a total of 32 crystallographic point groups, 30 of which are relevant to chemistry. Their classification is based on the Schoenflies notation.

Group theory A set of symmetry operations form a group, with operator the application of the operations itself, when: • • • •

the result of consecutive application (composition) of any two operations is also a member of the group (closure). the application of the operations is associative: A(BC) = AB(C) the group contains the identity operation, denoted E, such that AE = EA = A for any operation A in the group. For every operation A in the group, there is an inverse element A-1 in the group, for which AA-1 = A-1A = E

The order of a group is the number of symmetry operations for that group. For example, the point group for the water molecule is C2v, with symmetry operations E, C2, σv and σv'. Its order is thus 4. Each operation is its own inverse. As an example of closure, a C2 rotation followed by a σv reflection is seen to be a σv' symmetry operation: σv*C2 = σv'. (Note that "Operation A followed by B to form C" is written BA = C). Another example is the ammonia molecule, which is pyramidal and contains a three-fold rotation axis as well as three mirror planes at an angle of 120° to each other. Each mirror plane contains an N-H bond and bisects the H-N-H bond angle opposite to that bond. Thus ammonia molecule belongs to the C3v point group which has order 6: an identity element E, two rotation operations C3 and C32, and three mirror reflections σv, σv' and σv".

Molecular symmetry

195

Common point groups The following table contains a list of point groups with representative molecules. The description of structure includes common shapes of molecules based on VSEPR theory. Point group

Symmetry elements

Simple description, chiral if applicable

Illustrative species

C1

E

no symmetry, chiral

CFClBrH, lysergic acid

Cs

E σh

planar, no other symmetry

thionyl chloride, hypochlorous acid

Ci

Ei

Inversion center

anti-1,2-dichloro-1,2-dibromoethane

C∞v

E 2C∞ σv

linear

hydrogen chloride, dicarbon monoxide

D∞h

E 2C∞ ∞σi i 2S∞ ∞C2

linear with inversion center

dihydrogen, azide anion, carbon dioxide

C2

E C2

"open book geometry," chiral

hydrogen peroxide

C3

E C3

propeller, chiral

triphenylphosphine

C2h

E C2 i σh

planar with inversion center

trans-1,2-dichloroethylene

C3h

E C3 C32 σh S3 S35

propeller

Boric acid

C2v

E C2 σv(xz) σv'(yz)

angular (H2O) or see-saw (SF4)

water, sulfur tetrafluoride, sulfuryl fluoride

C3v

E 2C3 3σv

trigonal pyramidal

ammonia, phosphorus oxychloride

C4v

E 2C4 C2 2σv 2σd

square pyramidal

xenon oxytetrafluoride

D3

E C3(z) 3C2

triple helix, chiral

Tris(ethylenediamine)cobalt(III) cation

D2h

E C2(z) C2(y) C2(x) i σ(xy) σ(xz) σ(yz)

planar with inversion center

ethylene, dinitrogen tetroxide, diborane

D3h

E 2C3 3C2 σh 2S3 3σv

trigonal planar or trigonal bipyramidal

boron trifluoride, phosphorus pentachloride

D4h

E 2C4 C2 2C2' 2C2 i 2S4 σh 2σv 2σd

square planar

xenon tetrafluoride

D5h

E 2C5 2C52 5C2 σh 2S5 2S53 5σv

pentagonal

ruthenocene, eclipsed ferrocene, C70 fullerene

D6h

E 2C6 2C3 C2 3C2' 3C2 i 3S3 2S63 σh 3σd 3σv

hexagonal

benzene, bis(benzene)chromium

D2d

E 2S4 C2 2Ch 2C2' 2σd

90° twist

allene, tetrasulfur tetranitride

D3d

E 2C3 3C2 i 2S6 3σd

60° twist

ethane (staggered rotamer), cyclohexane (chair conformer)

D4d

E 2S8 2C4 2S83 C2 4C2' 4σd

45° twist

dimanganese decacarbonyl (staggered rotamer)

D5d

E 2C5 2C52 5C2 i 3S103 2S10 5σd

36° twist

ferrocene (staggered rotamer)

Td

E 8C3 3C2 6S4 6σd

tetrahedral

methane, phosphorus pentoxide, adamantane

Oh

E 8C3 6C2 6C4 3C2 i 6S4 8S6 3σh 6σd

octahedral or cubic

cubane, sulfur hexafluoride

Ih

E 12C5 12C52 20C3 15C2 i 12S10 12S103 20S6 15σ

icosahedral

C60, B12H122-

Molecular symmetry

Representations The symmetry operations can be represented in many ways. A convenient representation is by matrices. For any vector representing a point in Cartesian coordinates, left-multiplying it gives the new location of the point transformed by the symmetry operation. Composition of operations corresponds to matrix multiplication. In the C2v example this is:

Although an infinite number of such representations exist, the irreducible representations (or "irreps") of the group are commonly used, as all other representations of the group can be described as a linear combination of the irreducible representations.

Character tables For each point group, a character table summarizes information on its symmetry operations and on its irreducible representations. As there are always equal numbers of irreducible representations and classes of symmetry operations, the tables are square. The table itself consists of characters which represent how a particular irreducible representation transforms when a particular symmetry operation is applied. Any symmetry operation in a molecule's point group acting on the molecule itself will leave it unchanged. But for acting on a general entity, such as a vector or an orbital, this need not be the case. The vector could change sign or direction, and the orbital could change type. For simple point groups, the values are either 1 or −1: 1 means that the sign or phase (of the vector or orbital) is unchanged by the symmetry operation (symmetric) and −1 denotes a sign change (asymmetric). The representations are labeled according to a set of conventions: • A, when rotation around the principal axis is symmetrical • B, when rotation around the principal axis is asymmetrical • E and T are doubly and triply degenerate representations, respectively • when the point group has an inversion center, the subscript g (German: gerade or even) signals no change in sign, and the subscript u (ungerade or uneven) a change in sign, with respect to inversion. • with point groups C∞v and D∞h the symbols are borrowed from angular momentum description: Σ, Π, Δ. The tables also capture information about how the Cartesian basis vectors, rotations about them, and quadratic functions of them transform by the symmetry operations of the group, by noting which irreducible representation transforms in the same way. These indications are conventionally on the right hand side of the tables. This information is useful because chemically important orbitals (in particular p and d orbitals) have the same symmetries as these entities. The character table for the C2v symmetry point group is given below:

196

Molecular symmetry

197

C2v

E

C2 σv(xz) σv'(yz)

A1

1

1

1

1

z

x2, y2, z2

A2

1

1

−1

−1

Rz

xy

B1

1

−1 1

−1

x, Ry xz

B2

1

−1 −1

1

y, Rx yz

Consider the example of water (H2O) which has the C2v symmetry described above. The 2px orbital of oxygen is oriented perpendicular to the plane of the molecule and switches sign with a C2 and a σv'(yz) operation, but remains unchanged with the other two operations (obviously, the character for the identity operation is always +1). This orbital's character set is thus {1, −1, 1, −1}, corresponding to the B1 irreducible representation. Similarly, the 2pz orbital is seen to have the symmetry of the A1 irreducible representation, 2py B2, and the 3dxy orbital A2. These assignments and others are noted in the rightmost two columns of the table.

Historical background Hans Bethe used characters of point group operations in his study of ligand field theory in 1929, and Eugene Wigner used group theory to explain the selection rules of atomic spectroscopy[7] . The first character tables were compiled by László Tisza (1933), in connection to vibrational spectra. Robert Mulliken was the first to publish character tables in English (1933), and E. Bright Wilson used them in 1934 to predict the symmetry of vibrational normal modes.[8] The complete set of 32 crystallographic point groups was published in 1936 by Rosenthal and Murphy.[9]

References [1] [2] [3] [4] [5] [6]

Quantum Chemistry, Third Edition John P. Lowe, Kirk Peterson ISBN 0124575510 Physical Chemistry: A Molecular Approach by Donald A. McQuarrie, John D. Simon ISBN 0935702997 The chemical bond 2nd Ed. J.N. Murrell, S.F.A. Kettle, J.M. Tedder ISBN 0471907600 Physical Chemistry P. W. Atkins ISBN 0716728710 G. L. Miessler and D. A. Tarr “Inorganic Chemistry” 3rd Ed, Pearson/Prentice Hall publisher, ISBN 0-13-035471-6. LEO Ergebnisse für "einheit" (http:/ / dict. leo. org/ ende?lp=ende& lang=de& searchLoc=0& cmpType=relaxed& sectHdr=on& spellToler=on& search=einheit& relink=on) [7] Group Theory and its application to the quantum mechanics of atomic spectra, E. P. Wigner, Academic Press Inc. (1959) [8] Correcting Two Long-Standing Errors in Point Group Symmetry Character Tables Randall B. Shirts J. Chem. Educ. 2007, 84, 1882. Abstract (http:/ / jchemed. chem. wisc. edu/ Journal/ Issues/ 2007/ Nov/ abs1882. html) [9] Group Theory and the Vibrations of Polyatomic Molecules Jenny E. Rosenthal and G. M. Murphy Rev. Mod. Phys. 8, 317 - 346 (1936) doi:10.1103/RevModPhys.8.317

External links • Molecular symmetry @ University of Exeter Link (http://www.phys.ncl.ac.uk/staff/njpg/symmetry/) • Molecular symmetry @ Imperial College London Link (http://www.ch.ic.ac.uk/local/symmetry/) • Molecular Point Group Symmetry Tables (http://www.webqc.org/symmetry.php)

Applications of group theory

Applications of group theory In mathematics and abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have strongly influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced tremendous advances and have become subject areas in their own right. Various physical systems, such as crystals and the hydrogen atom, can be modelled by symmetry groups. Thus group theory and the closely related representation theory have many applications in physics and chemistry. One of the most important mathematical achievements of the 20th century was the collaborative effort, taking up more than 10,000 journal pages and mostly published between 1960 and 1980, that culminated in a complete classification of finite simple groups.

History Group theory has three main historical sources: number theory, the theory of algebraic equations, and geometry. The number-theoretic strand was begun by Leonhard Euler, and developed by Gauss's work on modular arithmetic and additive and multiplicative groups related to quadratic fields. Early results about permutation groups were obtained by Lagrange, Ruffini, and Abel in their quest for general solutions of polynomial equations of high degree. Évariste Galois coined the term “group” and established a connection, now known as Galois theory, between the nascent theory of groups and field theory. In geometry, groups first became important in projective geometry and, later, non-Euclidean geometry. Felix Klein's Erlangen program famously proclaimed group theory to be the organizing principle of geometry. Galois, in the 1830s, was the first to employ groups to determine the solvability of polynomial equations. Arthur Cayley and Augustin Louis Cauchy pushed these investigations further by creating the theory of permutation group. The second historical source for groups stems from geometrical situations. In an attempt to come to grips with possible geometries (such as euclidean, hyperbolic or projective geometry) using group theory, Felix Klein initiated the Erlangen programme. Sophus Lie, in 1884, started using groups (now called Lie groups) attached to analytic problems. Thirdly, groups were (first implicitly and later explicitly) used in algebraic number theory. The different scope of these early sources resulted in different notions of groups. The theory of groups was unified starting around 1880. Since then, the impact of group theory has been ever growing, giving rise to the birth of abstract algebra in the early 20th century, representation theory, and many more influential spin-off domains. The classification of finite simple groups is a vast body of work from the mid 20th century, classifying all the finite simple groups.

Main classes of groups The range of groups being considered has gradually expanded from finite permutation groups and special examples of matrix groups to abstract groups that may be specified through a presentation by generators and relations.

Permutation groups The first class of groups to undergo a systematic study was permutation groups. Given any set X and a collection G of bijections of X into itself (known as permutations) that is closed under compositions and inverses, G is a group acting on X. If X consists of n elements and G consists of all permutations, G is the symmetric group Sn; in general, G is a subgroup of the symmetric group of X. An early construction due to Cayley exhibited any group as a permutation group, acting on itself (X = G) by means of the left regular representation.

198

Applications of group theory In many cases, the structure of a permutation group can be studied using the properties of its action on the corresponding set. For example, in this way one proves that for n ≥ 5, the alternating group An is simple, i.e. does not admit any proper normal subgroups. This fact plays a key role in the impossibility of solving a general algebraic equation of degree n ≥ 5 in radicals.

Matrix groups The next important class of groups is given by matrix groups, or linear groups. Here G is a set consisting of invertible matrices of given order n over a field K that is closed under the products and inverses. Such a group acts on the n-dimensional vector space Kn by linear transformations. This action makes matrix groups conceptually similar to permutation groups, and geometry of the action may be usefully exploited to establish properties of the group G.

Transformation groups Permutation groups and matrix groups are special cases of transformation groups: groups that act on a certain space X preserving its inherent structure. In the case of permutation groups, X is a set; for matrix groups, X is a vector space. The concept of a transformation group is closely related with the concept of a symmetry group: transformation groups frequently consist of all transformations that preserve a certain structure. The theory of transformation groups forms a bridge connecting group theory with differential geometry. A long line of research, originating with Lie and Klein, considers group actions on manifolds by homeomorphisms or diffeomorphisms. The groups themselves may be discrete or continuous.

Abstract groups Most groups considered in the first stage of the development of group theory were "concrete", having been realized through numbers, permutations, or matrices. It was not until the late nineteenth century that the idea of an abstract group as a set with operations satisfying a certain system of axioms began to take hold. A typical way of specifying an abstract group is through a presentation by generators and relations,

A significant source of abstract groups is given by the construction of a factor group, or quotient group, G/H, of a group G by a normal subgroup H. Class groups of algebraic number fields were among the earliest examples of factor groups, of much interest in number theory. If a group G is a permutation group on a set X, the factor group G/H is no longer acting on X; but the idea of an abstract group permits one not to worry about this discrepancy. The change of perspective from concrete to abstract groups makes it natural to consider properties of groups that are independent of a particular realization, or in modern language, invariant under isomorphism, as well as the classes of group with a given such property: finite groups, periodic groups, simple groups, solvable groups, and so on. Rather than exploring properties of an individual group, one seeks to establish results that apply to a whole class of groups. The new paradigm was of paramount importance for the development of mathematics: it foreshadowed the creation of abstract algebra in the works of Hilbert, Emil Artin, Emmy Noether, and mathematicians of their school.

199

Applications of group theory

Topological and algebraic groups An important elaboration of the concept of a group occurs if G is endowed with additional structure, notably, of a topological space, differentiable manifold, or algebraic variety. If the group operations m (multiplication) and i (inversion),

are compatible with this structure, i.e. are continuous, smooth or regular (in the sense of algebraic geometry) maps then G becomes a topological group, a Lie group, or an algebraic group.[1] The presence of extra structure relates these types of groups with other mathematical disciplines and means that more tools are available in their study. Topological groups form a natural domain for abstract harmonic analysis, whereas Lie groups (frequently realized as transformation groups) are the mainstays of differential geometry and unitary representation theory. Certain classification questions that cannot be solved in general can be approached and resolved for special subclasses of groups. Thus, compact connected Lie groups have been completely classified. There is a fruitful relation between infinite abstract groups and topological groups: whenever a group Γ can be realized as a lattice in a topological group G, the geometry and analysis pertaining to G yield important results about Γ. A comparatively recent trend in the theory of finite groups exploits their connections with compact topological groups (profinite groups): for example, a single p-adic analytic group G has a family of quotients which are finite p-groups of various orders, and properties of G translate into the properties of its finite quotients.

Combinatorial and geometric group theory Groups can be described in different ways. Finite groups can be described by writing down the group table consisting of all possible multiplications g • h. A more important way of defining a group is by generators and relations, also called the presentation of a group. Given any set F of generators {gi}i ∈ I, the free group generated by F surjects onto the group G. The kernel of this map is called subgroup of relations, generated by some subset D. The presentation is usually denoted by 〈F | D 〉. For example, the group Z = 〈a | 〉 can be generated by one element a (equal to +1 or −1) and no relations, because n·1 never equals 0 unless n is zero. A string consisting of generator symbols is called a word. Combinatorial group theory studies groups from the perspective of generators and relations.[2] It is particularly useful where finiteness assumptions are satisfied, for example finitely generated groups, or finitely presented groups (i.e. in addition the relations are finite). The area makes use of the connection of graphs via their fundamental groups. For example, one can show that every subgroup of a free group is free. There are several natural questions arising from giving a group by its presentation. The word problem asks whether two words are effectively the same group element. By relating the problem to Turing machines, one can show that there is in general no algorithm solving this task. An equally difficult problem is, whether two groups given by different presentations are actually isomorphic. For example Z can also be presented by 〈x, y | xyxyx = 1〉 and it is not obvious (but true) that this presentation is isomorphic to the standard one above.

200

Applications of group theory

Geometric group theory attacks these problems from a geometric viewpoint, either by viewing groups as geometric objects, or by finding suitable geometric objects a group acts on.[3] The first idea is made precise by means of the Cayley graph, whose vertices correspond to group elements and edges correspond to right multiplication in the group. Given two elements, one constructs the word metric given by the length of the minimal path between the elements. A theorem of Milnor and Svarc then says that given a group G acting in a reasonable manner on a metric space X, for example a compact manifold, then G is quasi-isometric (i.e. looks similar from the far) to the space X.

201

The Cayley graph of 〈 x, y ∣ 〉, the free group of rank 2.

Representation of groups Saying that a group G acts on a set X means that every element defines a bijective map on a set in a way compatible with the group structure. When X has more structure, it is useful to restrict this notion further: a representation of G on a vector space V is a group homomorphism: ρ : G → GL(V), where GL(V) consists of the invertible linear transformations of V. In other words, to every group element g is assigned an automorphism ρ(g) such that ρ(g) ∘ ρ(h) = ρ(gh) for any h in G. This definition can be understood in two directions, both of which give rise to whole new domains of mathematics.[4] On the one hand, it may yield new information about the group G: often, the group operation in G is abstractly given, but via ρ, it corresponds to the multiplication of matrices, which is very explicit.[5] On the other hand, given a well-understood group acting on a complicated object, this simplifies the study of the object in question. For example, if G is finite, it is known that V above decomposes into irreducible parts. These parts in turn are much more easily manageable than the whole V (via Schur's lemma). Given a group G, representation theory then asks what representations of G exist. There are several settings, and the employed methods and obtained results are rather different in every case: representation theory of finite groups and representations of Lie groups are two main subdomains of the theory. The totality of representations is governed by the group's characters. For example, Fourier polynomials can be interpreted as the characters of U(1), the group of complex numbers of absolute value 1, acting on the L2-space of periodic functions.

Connection of groups and symmetry Given a structured object X of any sort, a symmetry is a mapping of the object onto itself which preserves the structure. This occurs in many cases, for example 1. If X is a set with no additional structure, a symmetry is a bijective map from the set to itself, giving rise to permutation groups. 2. If the object X is a set of points in the plane with its metric structure or any other metric space, a symmetry is a bijection of the set to itself which preserves the distance between each pair of points (an isometry). The corresponding group is called isometry group of X. 3. If instead angles are preserved, one speaks of conformal maps. Conformal maps give rise to Kleinian groups, for example. 4. Symmetries are not restricted to geometrical objects, but include algebraic objects as well. For instance, the equation

Applications of group theory

has the two solutions

202

, and

. In this case, the group that exchanges the two roots is the Galois

group belonging to the equation. Every polynomial equation in one variable has a Galois group, that is a certain permutation group on its roots. The axioms of a group formalize the essential aspects of symmetry. Symmetries form a group: they are closed because if you take a symmetry of an object, and then apply another symmetry, the result will still be a symmetry. The identity keeping the object fixed is always a symmetry of an object. Existence of inverses is guaranteed by undoing the symmetry and the associativity comes from the fact that symmetries are functions on a space, and composition of functions are associative. Frucht's theorem says that every group is the symmetry group of some graph. So every abstract group is actually the symmetries of some explicit object. The saying of "preserving the structure" of an object can be made precise by working in a category. Maps preserving the structure are then the morphisms, and the symmetry group is the automorphism group of the object in question.

Applications of group theory Applications of group theory abound. Almost all structures in abstract algebra are special cases of groups. Rings, for example, can be viewed as abelian groups (corresponding to addition) together with a second operation (corresponding to multiplication). Therefore group theoretic arguments underlie large parts of the theory of those entities. Galois theory uses groups to describe the symmetries of the roots of a polynomial (or more precisely the automorphisms of the algebras generated by these roots). The fundamental theorem of Galois theory provides a link between algebraic field extensions and group theory. It gives an effective criterion for the solvability of polynomial equations in terms of the solvability of the corresponding Galois group. For example, S5, the symmetric group in 5 elements, is not solvable which implies that the general quintic equation cannot be solved by radicals in the way equations of lower degree can. The theory, being one of the historical roots of group theory, is still fruitfully applied to yield new results in areas such as class field theory. Algebraic topology is another domain which prominently associates groups to the objects the theory is interested in. There, groups are used to describe certain invariants of topological spaces. They are called "invariants" because they are defined in such a way that they do not change if the space is subjected to some deformation. For example, the fundamental group "counts" how many paths in the space are essentially different. The Poincaré conjecture, proved in 2002/2003 by Grigori Perelman is a prominent application of this idea. The influence is not unidirectional, though. For example, algebraic topology makes use of Eilenberg–MacLane spaces which are spaces with prescribed homotopy groups. Similarly algebraic K-theory stakes in a crucial way on classifying spaces of groups. Finally, the name of the torsion subgroup of an infinite group shows the legacy of topology in group theory. Algebraic geometry and cryptography likewise uses group theory in many ways. Abelian varieties have been introduced above. The presence of the group operation yields additional information which makes these varieties particularly accessible. They also often serve as a test for new conjectures.[6] The one-dimensional case, namely elliptic curves is studied in particular detail. They are both theoretically and practically intriguing.[7] Very large groups of prime order constructed in Elliptic-Curve Cryptography serve for public key cryptography. Cryptographical methods of this kind

A torus. Its abelian group structure is induced from the map C → C/Z+τZ, where τ is a parameter.

Applications of group theory

The cyclic group Z26 underlies Caesar's cipher.

203 benefit from the flexibility of the geometric objects, hence their group structures, together with the complicated structure of these groups, which make the discrete logarithm very hard to calculate. One of the earliest encryption protocols, Caesar's cipher, may also be interpreted as a (very easy) group operation. In another direction, toric varieties are algebraic varieties acted on by a torus. Toroidal embeddings have recently led to advances in algebraic geometry, in particular resolution of singularities.[8]

Algebraic number theory is a special case of group theory, thereby following the rules of the latter. For example, Euler's product formula

captures the fact that any integer decomposes in a unique way into primes. The failure of this statement for more general rings gives rise to class groups and regular primes, which feature in Kummer's treatment of Fermat's Last Theorem. • The concept of the Lie group (named after mathematician Sophus Lie) is important in the study of differential equations and manifolds; they describe the symmetries of continuous geometric and analytical structures. Analysis on these and other groups is called harmonic analysis. Haar measures, that is integrals invariant under the translation in a Lie group, are used for pattern recognition and other image processing techniques.[9] • In combinatorics, the notion of permutation group and the concept of group action are often used to simplify the counting of a set of objects; see in particular Burnside's lemma. • The presence of the 12-periodicity in the circle of fifths yields applications of elementary group theory in musical set theory. • In physics, groups are important because they describe the symmetries which the laws of physics seem to obey. Physicists are very interested in group representations, especially of Lie groups, since these representations often point the way to the "possible" physical theories. Examples of the use of groups in physics include the Standard Model, gauge theory, the Lorentz group, and the Poincaré group. • In chemistry and materials science, groups are used to classify crystal structures, regular polyhedra, and the symmetries of molecules. The assigned point groups can then be used to determine physical properties (such as polarity and chirality), spectroscopic properties (particularly useful for Raman spectroscopy and infrared spectroscopy), and to construct molecular orbitals.

The circle of fifths may be endowed with a cyclic group structure

Applications of group theory

See also • Group (mathematics) • Glossary of group theory • List of group theory topics

Notes [1] This process of imposing extra structure has been formalized through the notion of a group object in a suitable category. Thus Lie groups are group objects in the category of differentiable manifolds and affine algebraic groups are group objects in the category of affine algebraic varieties. [2] Schupp & Lyndon 2001 [3] La Harpe 2000 [4] Such as group cohomology or equivariant K-theory. [5] In particular, if the representation is faithful. [6] For example the Hodge conjecture (in certain cases). [7] See the Birch-Swinnerton-Dyer conjecture, one of the millennium problems [8] Abramovich, Dan; Karu, Kalle; Matsuki, Kenji; Wlodarczyk, Jaroslaw (2002), "Torification and factorization of birational maps", Journal of the American Mathematical Society 15 (3): 531–572, doi:10.1090/S0894-0347-02-00396-X, MR1896232 [9] Lenz, Reiner (1990), Group theoretical methods in image processing (http:/ / webstaff. itn. liu. se/ ~reile/ LNCS413/ index. htm), Lecture Notes in Computer Science, 413, Berlin, New York: Springer-Verlag, doi:10.1007/3-540-52290-5, ISBN 978-0-387-52290-6,

References • Borel, Armand (1991), Linear algebraic groups, Graduate Texts in Mathematics, 126 (2nd ed.), Berlin, New York: Springer-Verlag, MR1102012, ISBN 978-0-387-97370-8 • Carter, Nathan C. (2009), Visual group theory (http://web.bentley.edu/empl/c/ncarter/vgt/), Classroom Resource Materials Series, Mathematical Association of America, MR2504193, ISBN 978-0-88385-757-1 • Cannon, John J. (1969), "Computers in group theory: A survey", Communications of the Association for Computing Machinery 12: 3–12, doi:10.1145/362835.362837, MR0290613 • Frucht, R. (1939), "Herstellung von Graphen mit vorgegebener abstrakter Gruppe" (http://www.numdam.org/ numdam-bin/fitem?id=CM_1939__6__239_0), Compositio Mathematica 6: 239–50, ISSN 0010-437X • Golubitsky, Martin; Stewart, Ian (2006), "Nonlinear dynamics of networks: the groupoid formalism", Bull. Amer. Math. Soc. (N.S.) 43: 305–364, doi:10.1090/S0273-0979-06-01108-6, MR2223010 Shows the advantage of generalising from group to groupoid. • Judson, Thomas W. (1997), Abstract Algebra: Theory and Applications (http://abstract.ups.edu) An introductory undergraduate text in the spirit of texts by Gallian or Herstein, covering groups, rings, integral domains, fields and Galois theory. Free downloadable PDF with open-source GFDL license. • Kleiner, Israel (1986), "The evolution of group theory: a brief survey" (http://jstor.org/stable/2690312), Mathematics Magazine 59 (4): 195–215, doi:10.2307/2690312, MR863090, ISSN 0025-570X • La Harpe, Pierre de (2000), Topics in geometric group theory, University of Chicago Press, ISBN 978-0-226-31721-2 • Livio, M. (2005), The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry, Simon & Schuster, ISBN 0-7432-5820-7 Conveys the practical value of group theory by explaining how it points to symmetries in physics and other sciences. • Mumford, David (1970), Abelian varieties, Oxford University Press, ISBN 978-0-19-560528-0, OCLC 138290 • Ronan M., 2006. Symmetry and the Monster. Oxford University Press. ISBN 0-19-280722-6. For lay readers. Describes the quest to find the basic building blocks for finite groups. • Rotman, Joseph (1994), An introduction to the theory of groups, New York: Springer-Verlag, ISBN 0-387-94285-8 A standard contemporary reference.

204

Applications of group theory • Schupp, Paul E.; Lyndon, Roger C. (2001), Combinatorial group theory, Berlin, New York: Springer-Verlag, ISBN 978-3-540-41158-1 • Scott, W. R. (1987) [1964], Group Theory, New York: Dover, ISBN 0-486-65377-3 Inexpensive and fairly readable, but somewhat dated in emphasis, style, and notation. • Shatz, Stephen S. (1972), Profinite groups, arithmetic, and geometry, Princeton University Press, MR0347778, ISBN 978-0-691-08017-8 • Weibel, Charles A. (1994), An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, 38, Cambridge University Press, MR1269324, ISBN 978-0-521-55987-4, OCLC 36131259

External links • History of the abstract group concept (http://www-history.mcs.st-andrews.ac.uk/history/HistTopics/ Abstract_groups.html) • Higher dimensional group theory (http://www.bangor.ac.uk/r.brown/hdaweb2.htm) This presents a view of group theory as level one of a theory which extends in all dimensions, and has applications in homotopy theory and to higher dimensional nonabelian methods for local-to-global problems. • Plus teacher and student package: Group Theory (http://plus.maths.org/issue48/package/index.html) This package brings together all the articles on group theory from Plus, the online mathematics magazine produced by the Millennium Mathematics Project at the University of Cambridge, exploring applications and recent breakthroughs, and giving explicit definitions and examples of groups. • US Naval Academy group theory guide (http://www.usna.edu/Users/math/wdj/tonybook/gpthry/node1. html) A general introduction to group theory with exercises written by Tony Gaglione.

Examples of groups Some elementary examples of groups in mathematics are given on Group (mathematics). Further examples are listed here.

Permutations of a set of three elements

205

Examples of groups

206

Consider three colored blocks (red, green, and blue), initially placed in the order RGB. Let a be the operation "swap the first block and the second block", and b be the operation "swap the second block and the third block". We can write xy for the operation "first do y, then do x"; so that ab is the operation RGB → RBG → BRG, which could be described as "move the first two blocks one position to the right and put the third block into the first position". If we write e for "leave the blocks as they are" (the identity operation), then we can write the six permutations of the three blocks as follows: • e : RGB → RGB • a : RGB → GRB • b : RGB → RBG • ab : RGB → BRG • ba : RGB → GBR • aba : RGB → BGR

Cycle graph for S3 (or D6). A loop specifies a series of powers of any element connected to the identity element (1). For example, the e-ba-ab loop reflects the fact that ba2=ab and ba3=e, as well as the fact that ab2=ba and ab3=e The other "loops" are roots of unity so that, for example a2=e.

Note that aa has the effect RGB → GRB → RGB; so we can write aa = e. Similarly, bb = (aba)(aba) = e; (ab)(ba) = (ba)(ab) = e; so every element has an inverse. By inspection, we can determine associativity and closure; note in particular that (ba)b = aba = b(ab). Since it is built up from the basic operations a and b, we say that the set {a,b} generates this group. The group, called the symmetric group S3, has order 6, and is non-abelian (since, for example, ab ≠ ba).

The group of translations of the plane A translation of the plane is a rigid movement of every point of the plane for a certain distance in a certain direction. For instance "move in the North-East direction for 2 miles" is a translation of the plane. If you have two such translations a and b, they can be composed to form a new translation a ∘ b as follows: first follow the prescription of b, then that of a. For instance, if a = "move North-East for 3 miles" and b = "move South-East for 4 miles" then a ∘ b = "move East for 5 miles" (see Pythagorean theorem for why this is so, geometrically). The set of all translations of the plane with composition as operation forms a group: 1. If a and b are translations, then a ∘ b is also a translation. 2. Composition of translations is associative: (a ∘ b) ∘ c = a ∘ (b ∘ c). 3. The identity element for this group is the translation with prescription "move zero miles in whatever direction you like". 4. The inverse of a translation is given by walking in the opposite direction for the same distance. This is an Abelian group and our first (nondiscrete) example of a Lie group: a group which is also a manifold.

Examples of groups

The symmetry group of a square - dihedral group of order 8 Groups are very important to describe the symmetry of objects, be they geometrical (like a tetrahedron) or algebraic (like a set of equations). As an example, we consider a square concrete slab of a certain thickness. In order to describe its symmetry, we form the set of all those rigid movements of the slab that don't make a visible difference. For instance, if you turn it by 90 degrees clockwise, then it still looks the same, so this movement is one element of our set, let's call it R. We could also flip the slab horizontally so that its underside become up. Again, after performing this movement, the slab looks the same, so this is also an element of our set and we call it T. Then there's of course the movement that does nothing; it's denoted by I. Now if you have two such movements a and b, you can define the composition a ∘ b as above: you first perform the movement b and then the movement a. The result will leave the slab looking like before. The point is that the set of all those movements, with composition as operation, forms a group. This group is the most concise description of the slab's symmetry. Chemists use symmetry groups of this type to describe the symmetry of crystals. Let's investigate our slab symmetry group some more. Right now, we have the elements R, T and I, but we can easily form more: for instance R ∘ R, also written as R2, is a 180 degree turn (clockwise or counter-clockwise doesn't matter). R3 is a 270 degree clockwise rotation, or, what is the same thing, a 90 degree counter-clockwise rotation. We also see that T2 = I and also R4 = I. Here's an interesting one: what does R ∘ T do? First flip horizontally, then rotate. Try to visualize that R ∘ T = T ∘ R3. Also, R2 ∘ T is a vertical flip and is equal to T ∘ R2. This group is finite with order 8 and has Cayley table

Cycle graph for D4. A loop specifies a series of powers of any element connected to the identity element (e). For example, the e-a-a2-a3 loop reflects the fact that the successive powers of a are distinct until a4=e. This loop also reflects the fact that successive powers of a3 are distinct until (a3)4=e. The other "loops" are roots of the identity so that, for example b2=e. In the text of the article, R=a, T=b and I=e.

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Examples of groups

208

∘

I

T

R

R2

R3

R'T R2T R3T

I

I

T

R

R2

R3

R'T R2T R3T

T

T

I

R

R

R2 R3

R3T R2T R'T

R3

R2

R

R'T

R2

R3

I

R2T R3T

T

R2

R2T

R3

I

R

R3T

R3

R3T

I

R

R2

T

R'T R2T

R'T R'T

R

T

R3T R2T

I

R3

R2

R2T R2T

R2

R'T

R3T

R

I

R3

R3T R3T

R3

R2T R'T

T

R2

R

I

T

T

R'T

For any two elements in the group, the table records what their composition is. Note how every element appears in every row and every column exactly once; this is not a coincidence. You may want to verify some entries. Here we wrote "R3T" as a short hand for R3 ∘ T. Mathematicians know this group as the dihedral group of order 8, and call it either D4 or D8 depending on what notation they use for dihedral groups. This was an example of a non-abelian group: the operation ∘ here is not commutative, which you can see from the table; the table is not symmetrical about the main diagonal. The dihedral group of order 8 is isomorphic to the permutation group generated by (1234) and (13).

Matrix groups If n is some positive integer, we can consider the set of all invertible n by n matrices over the reals, say. This is a group with matrix multiplication as operation. It is called the general linear group, GL(n). Geometrically, it contains all combinations of rotations, reflections, dilations and skew transformations of n-dimensional Euclidean space that fix a given point (the origin). If we restrict ourselves to matrices with determinant 1, then we get another group, the special linear group, SL(n). Geometrically, this consists of all the elements of GL(n) that preserve both orientation and volume of the various geometric solids in Euclidean space. If instead we restrict ourselves to orthogonal matrices, then we get the orthogonal group O(n). Geometrically, this consists of all combinations of rotations and reflections that fix the origin. These are precisely the transformations which preserve lengths and angles. Finally, if we impose both restrictions, then we get the special orthogonal group SO(n), which consists of rotations only. These groups are our first examples of infinite non-abelian groups. They are also happen to be Lie groups. In fact, most of the important Lie groups (but not all) can be expressed as matrix groups. If this idea is generalised to matrices with complex numbers as entries, then we get further useful Lie groups, such as the unitary group U(n). We can also consider matrices with quaternions as entries; in this case, there is no well-defined notion of a determinant (and thus no good way to define a quaternionic "volume"), but we can still define a group analogous to the orthogonal group, the symplectic group Sp(n). Furthermore, the idea can be treated purely algebraically with matrices over any field, but then the groups are not Lie groups. For example, we have the general linear groups over finite fields. The group theorist J. L. Alperin has written that "The typical example of a finite group is GL(n,q), the general linear group of n dimensions over the field with q

Examples of groups elements. The student who is introduced to the subject with other examples is being completely misled." (Bulletin (New Series) of the American Mathematical Society, 10 (1984) 121)

Free group on two generators The free group with two generators a and b consists of all finite strings that can be formed from the four symbols a, a-1, b and b-1 such that no a appears directly next to an a-1 and no b appears directly next to an b-1. Two such strings can be concatenated and converted into a string of this type by repeatedly replacing the "forbidden" substrings with the empty string. For instance: "abab-1a-1" concatenated with "abab-1a" yields "abab-1a-1abab-1a", which gets reduced to "abaab-1a". One can check that the set of those strings with this operation forms a group with neutral element the empty string ε := "". (Usually the quotation marks are left off, which is why you need the symbol ε!) This is another infinite non-abelian group. Free groups are important in algebraic topology; the free group in two generators is also used for a proof of the Banach–Tarski paradox.

The set of maps The sets of maps from a set to a group Let G be a group and S a nonempty set. The set of maps M(S, G) is itself a group; namely for two maps f,g of S into G we define fg to be the map such that (fg)(x) = f(x)g(x) for every x∈S and f−1 to be the map such that f−1(x) = f(x)−1. Take maps f, g, and h in M(S,G). For every x in S, f(x) and g(x) are both in G, and so is (fg)(x). Therefore fg is also in M(S, G), or M(S, G) is closed. For ((fg)h)(x) = (fg)(x)h(x) = (f(x)g(x))h(x) = f(x)(g(x)h(x)) = f(x)(gh)(x) = (f(gh))(x), M(S, G) is associative. And there is a map i such that i(x) = e where e is the unit element of G. The map i makes all the functions f in M(S, G) such that if = fi = f, or i is the unit element of M(S, G). Thus, M(S, G) is actually a group. If G is commutative, then (fg)(x) = f(x)g(x) = g(x)f(x) = (gf)(x). Therefore so is M(S, G).

The groups of permutations Let G be the set of bijective mappings of a set S onto itself. Then G, also denoted by Perm(S) or Sym(S), is a group with ordinary composition of mappings. The unit element of G is the identity map of S. For two maps f and g in G are bijective, fg is also bijective. Therefore G is closed. The composition of maps is associative; hence G is a group. S may be either finite, or infinite.

Some more finite groups • list of small groups • List of the 230 crystallographic 3D space groups

209

Modular representation theory

Modular representation theory Modular representation theory is a branch of mathematics, and that part of representation theory that studies linear representations of finite group G over a field K of positive characteristic. As well as having applications to group theory, modular representations arise naturally in other branches of mathematics, such as algebraic geometry, coding theory, combinatorics and number theory. Within finite group theory, character-theoretic results proved by Richard Brauer using modular representation theory played an important role in early progress towards the classification of finite simple groups, especially for simple groups whose characterization was not amenable to purely group-theoretic methods because their Sylow 2 subgroups were too small in an appropriate sense. Also, a general result on embedding of elements of order 2 in finite groups called the Z* theorem, proved by George Glauberman using the theory developed by Brauer, was particularly useful in the classification program. If the characteristic of K does not divide the order of G, then modular representations are completely reducible, as with ordinary (characteristic 0) representations, by virtue of Maschke's theorem. The proof of Maschke's theorem relies on being able to divide by the group order, which is not meaningful when the order of G is divisible by the characteristic of K. In that case, representations need not be completely reducible, unlike the ordinary (and the coprime characteristic) case. Much of the discussion below implicitly assumes that the field K is sufficiently large (for example, K algebraically closed suffices), otherwise some statements need refinement.

History The earliest work on representation theory over finite fields is by Dickson (1902) who showed that when p does not divide the order of the group then the representation theory is similar to that in characteristic 0. He also investigated modular invariants of some finite groups. The systematic study of modular representations, when the characteristic divides the order of the group, was started by Brauer (1935) and continued by him for the next few decades.

Example Finding a representation of the cyclic group of two elements over F2 is equivalent to the problem of finding matrices whose square is the identity matrix. Over every field of characteristic other than 2, there is always a basis such that the matrix can be written as a diagonal matrix with only 1 or −1 occurring on the diagonal, such as

Over F2, there are many other possible matrices, such as

Over an algebraically closed field of positive characteristic, the representation theory of a finite cyclic group is fully explained by the theory of the Jordan normal form. Non-diagonal Jordan forms occur when the characteristic divides the order of the group.

210

Modular representation theory

Ring theory interpretation Given a field K and a finite group G, the group algebra K[G] (which is the K-vector space with K-basis consisting of the elements of G, endowed with algebra multiplication by extending the multiplication of G by linearity) is an Artinian ring. When the order of G is divisible by the characteristic of K, the group algebra is not semisimple, hence has non-zero Jacobson radical. In that case, there are finite-dimensional modules for the group algebra that are not projective modules. By contrast, in the characteristic 0 case every irreducible representation is a direct summand of the regular representation, hence is projective.

Brauer characters Modular representation theory was developed by Richard Brauer from about 1940 onwards to study in greater depth the relationships between the characteristic p representation theory, ordinary character theory and structure of G, especially as the latter relates to the embedding of, and relationships between, its p-subgroups. Such results can be applied in group theory to problems not directly phrased in terms of representations. Brauer introduced the notion now known as the Brauer character. When K is algebraically closed of positive characteristic p, there is a bijection between roots of unity in K and complex roots of unity of order prime to p. Once a choice of such a bijection is fixed, the Brauer character of a representation assigns to each group element of order coprime to p the sum of complex roots of unity corresponding to the eigenvalues (including multiplicities) of that element in the given representation. The Brauer character of a representation determines its composition factors but not, in general, its equivalence type. The irreducible Brauer characters are those afforded by the simple modules. These are integral ( though not necessarily non-negative) combinations of the restrictions to elements of order coprime to p of the ordinary irreducible characters. Conversely, the restriction to the elements of order prime to p of each ordinary irreducible character is uniquely expressible as a non-negative integer combination of irreducible Brauer characters.

Reduction (mod p) In the theory initially developed by Brauer, the link between ordinary representation theory and modular representation theory is best exemplified by considering the group algebra of the group G over a complete discrete valuation ring R with residue field K of positive characteristic p and field of fractions F of characteristic 0. The structure of R[G] is closely related both to the structure of the group algebra K[G] and to the structure of the semisimple group algebra F[G], and there is much interplay between the module theory of the three algebras. Each R[G]-module naturally gives rise to an F[G]-module, and, by a process often known informally as reduction (mod p), to a K[G]-module. On the other hand, since R is a principal ideal domain, each finite-dimensional F[G]-module arises by extension of scalars from an R[G]-module. In general, however, not all K[G]-modules arise as reductions (mod p) of R[G]-modules. Those that do are liftable.

211

Modular representation theory

Number of simple modules In ordinary representation theory, the number of simple modules k(G) is equal to the number of conjugacy classes of G. In the modular case, the number l(G) of simple modules is equal to the number of conjugacy classes whose elements have order coprime to the relevant prime p, the so-called p-regular classes.

Blocks and the structure of the group algebra In modular representation theory, while Maschke's theorem does not hold when the characteristic divides the group order, the group algebra may be decomposed as the direct sum of a maximal collection of two-sided ideals known as blocks (when the field K has characteristic 0, or characteristic coprime to the group order, there is also such a decomposition of the group algebra K[G] as a sum of blocks (one for each isomorphism type of simple module), but the situation is relatively transparent (at least when K is sufficiently large): each block is a full matrix algebra over K, the endomorphism ring of the vector space underlying the associated simple module). To obtain the blocks, the identity element of the group G is decomposed as a sum of primitive idempotents in Z(R[G]), the center of the group algebra over the maximal order R of F. The block corresponding to the primitive idempotent e is the two-sided ideal e R[G]. For each indecomposable R[G]-module, there only one such primitive idempotent that does not annihilate it, and the module is said to belong to (or to be in) the corresponding block (in which case, all its composition factors also belong to that block). In particular, each simple module belongs to a unique block. Each ordinary irreducible character may also be assigned to a unique block according to its decomposition as a sum of irreducible Brauer characters. The block containing the trivial module is known as the principal block.

Projective modules In ordinary representation theory, every indecomposable module is irreducible, and so every module is projective. However, the simple modules with characteristic dividing the group order are rarely projective. Indeed, if a simple module is projective, then it is the only simple module in its block, which is then isomorphic to the endomorphism algebra of the underlying vector space, a full matrix algebra. In that case, the block is said to have 'defect 0'. Generally, the structure of projective modules is difficult to determine. For the group algebra of a finite group, the (isomorphism types of) projective indecomposable modules are in a one-to-one correspondence with the (isomorphism types of) simple modules: the socle of each projective indecomposable is simple (and isomorphic to the top), and this affords the bijection, as non-isomorphic projective indecomposables have non-isomorphic socles. The multiplicity of a projective indecomposable module as a summand of the group algebra (viewed as the regular module) is the dimension of its socle (for large enough fields of characteristic zero, this recovers the fact that each simple module occurs with multiplicity equal to its dimension as a direct summand of the regular module). Each projective indecomposable module (and hence each projective module) in positive characteristic p may be lifted to a module in characteristic 0. Using the ring R as above, with residue field K, the identity element of G may be decomposed as a sum of mutually orthogonal primitive idempotents ( not necessarily central) of K[G]. Each projective indecomposable K[G]-module is isomorphic to e.K[G] for a primitive idempotent e that occurs in this decomposition. The idempotent e lifts to a primitive idempotent, say E, of R[G], and the left module E.R[G] has reduction (mod p) isomorphic to e.K[G].

212

Modular representation theory

Some orthogonality relations for Brauer characters When a projective module is lifted, the associated character vanishes on all elements of order divisible by p, and (with consistent choice of roots of unity), agrees with the Brauer character of the original characteristic p module on p-regular elements. The (usual character-ring) inner product of the Brauer character of a projective indecomposable with any other Brauer character can thus be defined: this is 0 if the second Brauer character is that of the socle of a non-isomorphic projective indecomposable, and 1 if the second Brauer character is that of its own socle. The multiplicity of an ordinary irreducible character in the character of the lift of a projective indecomposable is equal to the number of occurrences of the Brauer character of the socle of the projective indecomposable when the restriction of the ordinary character to p-regular elements is expressed as a sum of irreducible Brauer characters.

Decomposition matrix and Cartan matrix The composition factors of the projective indecomposable modules may be calculated as follows: Given the ordinary irreducible and irreducible Brauer characters of a particular finite group, the irreducible ordinary characters may be decomposed as non-negative integer combinations of the irreducible Brauer characters. The integers involved can be placed in a matrix, with the ordinary irreducible characters assigned rows and the irreducible Brauer characters assigned columns. This is referred to as the decomposition matrix, and is frequently labelled D. It is customary to place the trivial ordinary and Brauer characters in the first row and column respectively. The product of the transpose of D with D itself results in the Cartan matrix, usually denoted C; this is a symmetric matrix such that the entries in its j-th row are the multiplicities of the respective simple modules as composition factors of the j-th projective indecomposable module. The Cartan matrix is non-singular; in fact, its determinant is a power of the characteristic of K. Since a projective indecomposable module in a given block has all its composition factors in that same block, each block has its own Cartan matrix.

Defect groups To each block B of the group algebra K[G], Brauer associated a certain p-subgroup, known as its defect group (where p is the characteristic of K). Formally, it is the largest p-subgroup D of G for which there is a Brauer correspondent of B for the subgroup . The defect group of a block is unique up to conjugacy and has a strong influence on the structure of the block. For example, if the defect group is trivial, then the block contains just one simple module, just one ordinary character, the ordinary and Brauer irreducible characters agree on elements of order prime to the relevant characteristic p, and the simple module is projective. At the other extreme, when K has characteristic p, the Sylow p-subgroup of the finite group G is a defect group for the principal block of K[G]. The order of the defect group of a block has many arithmetical characterizations related to representation theory. It is the largest invariant factor of the Cartan matrix of the block, and occurs with multiplicity one. Also, the power of p dividing the index of the defect group of a block is the greatest common divisor of the powers of p dividing the dimensions of the simple modules in that block, and this coincides with the greatest common divisor of the powers of p dividing the degrees of the ordinary irreducible characters in that block. Other relationships between the defect group of a block and character theory include Brauer's result that if no conjugate of the p-part of a group element g is in the defect group of a given block, then each irreducible character in that block vanishes at g. This is a one of many consequences of Brauer's second main theorem. The defect group of a block also has several characterizations in the more module-theoretic approach to block theory, building on the work of J. A. Green, which associates a p-subgroup known as the vertex to an indecomposable module, defined in terms of relative projectivity of the module. For example, the vertex of each indecomposable module in a block is contained (up to conjugacy) in the defect group of the block, and no proper subgroup of the

213

Modular representation theory defect group has that property. Brauer's first main theorem states that the number of blocks of a finite group that have a given p-subgroup as defect group is the same as the corresponding number for the normalizer in the group of that p-subgroup. The easiest block structure to analyse with non-trivial defect group is when the latter is cyclic. Then there are only finitely many isomorphism types of indecomposable modules in the block, and the structure of the block is by now well understood, by virtue of work of Brauer, E.C. Dade, J.A.Green and J.G.Thompson, among others. In all other cases, there are infinitely many isomorphism types of indecomposable modules in the block. Blocks whose defect groups are not cyclic can be divided into two types: tame and wild. The tame blocks (which only occur for the prime 2) have as a defect group a dihedral group, semidihedral group or (generalized) quaternion group, and their structure has been broadly determined in a series of papers by Karin Erdmann. The indecomposable modules in wild blocks are extremely difficult to classify, even in principle.

References • Brauer, R. (1935), Über die Darstellung von Gruppen in Galoisschen Feldern [1], Actualités Scientifiques et Industrielles,, 195, Hermann et cie, Paris ., pp. 1–15, review [2] • Dickson, Leonard Eugene (1902), "On the Group Defined for any Given Field by the Multiplication Table of Any Given Finite Group" [3] (in English), Transactions of the American Mathematical Society (Providence, R.I.: American Mathematical Society) 3 (3): 285–301, ISSN 0002-9947 • Jean-Pierre Serre (1977). Linear Representations of Finite Groups. Springer-Verlag. ISBN 0-387-90190-6. • Walter Feit (1982). The representation theory of finite groups. North-Holland Mathematical Library. 25. Amsterdam-New York: North-Holland Publishing. ISBN 0-444-86155-6.

References [1] http:/ / books. google. com/ books?id=hkexAAAAIAAJ [2] http:/ / projecteuclid. org/ euclid. bams/ 1183499883 [3] http:/ / www. jstor. org/ stable/ 1986379

214

Conway group

Conway group In mathematics, the Conway groups Co1, Co2, and Co3 are three sporadic groups discovered by John Horton Conway. The largest of the Conway groups, Co1, of order 4,157,776,806,543,360,000, is obtained as the quotient of Co0 (automorphism group of Λ) by its center, which consists of the scalar matrices ±1. The groups Co2 (of order 42,305,421,312,000) and Co3 (of order 495,766,656,000) consist of the automorphisms of Λ fixing a lattice vector of type 2 and a vector of type 3 respectively. (The type of a vector is half of its square norm, v·v.) As the scalar −1 fixes no non-zero vector, these two groups are isomorphic to subgroups of Co1.

History Thomas Thompson relates how John Leech about 1964 investigated close packings of spheres in Euclidean spaces of large dimension. One of Leech's discoveries was a lattice packing in 24-space, based on what came to be called the Leech lattice Λ. He wondered whether his lattice's symmetry group contained an interesting simple group, but felt he needed the help of someone better acquainted with group theory. He had to do much asking around because the mathematicians were pre-occupied with agendas of their own. John Conway agreed to look at the problem. John G. Thompson said he would be interested if he were given the order of the group. Conway expected to spend months or years on the problem, but found results in just a few sessions.

Other sporadic groups Conway and Thompson found that 4 recently discovered sporadic simple groups were isomorphic to subgroups or quotients of subgroups of Co1. Two of these (subgroups of Co2 and Co3) can be defined as pointwise stabilizers of triangles with vertices, of sum zero, of types 2 and 3. A 2-2-3 triangle is fixed by the McLaughlin group McL (order 898,128,000). A 2-3-3 triangle is fixed by the Higman-Sims group (order 44,352,000). Two other sporadic groups can be defined as stabilizers of structures on the Leech lattice. Identifying R24 with C12 and Λ with Z[e2πi/3]12, the resulting automorphism group, i.e., the group of Leech lattice automorphisms preserving the complex structure, when divided by the 6-element group of complex scalar matrices, gives the Suzuki group Suz (of order 448,345,497,600). Suz is the only sporadic proper subquotient of Co1 that retains 13 as a prime factor. This group was discovered by Michio Suzuki in 1968. A similar construction gives the Hall-Janko group J2 (of order 604,800) as the quotient of the group of quaternionic automorphisms of Λ by the group ±1 of scalars. The 7 simple groups described above comprise what Robert Griess calls the second generation of the Happy Family, which consists of the 20 sporadic simple groups found within the Monster group. Several of the 7 groups contain at least some of the 5 Mathieu groups, which comprise the first generation. There was a conference on group theory held May 2-4, 1968, at Harvard University. Richard Brauer and Chih-Han Shah later published a book of its proceedings. It included important lectures on four groups of the second generation, but was a little too early to include the Conway groups. It has on the other hand been observed that if Conway had started a few years earlier, he could have discovered all 7 groups. Conway unified 4 seemingly rather unrelated groups into one larger group.

215

Conway group

216

An important maximal subgroup of Co0 Conway started his investigation with a subgroup called N. The Leech lattice is defined by use of the binary Golay code, whose automorphism group is the Mathieu group M24. Let E be a multiplicative representation of this code, a group of diagonal 24-by-24 matrices whose diagonal elements equal 1 or -1. E is an abelian group of type 212. Define N as the holomorph E:M24. Conway found that N is a maximal subgroup of Co0 and contains 2-Sylow subgroups of Co0. He used N to deduce the order of Co0. The negative of the identity is in E and commutes with every 24-by-24 matrix. Then Co1 has a maximal subgroup with structure 211:M24. The matrices of N have components that are integers. Since N is maximal in Co0 matrices in Co0.

[1]

, N is the group of all integral

Maximal subgroups of Co1 Co1 has 22 conjugacy classes of maximal subgroups. The maximal subgroups of Co1 are as follows. • Co2 • 3.Suz:2 (order divisible by 13) • 211:M24 • • • • • • • • • • • • • • • • • • •

Co3 21+8.O8+(2) U6(2):S3 (A4 × G2(4)):2 (order divisible by 13) 22+12:(A8 × S3) 24+12.(S3 × 3.S6) 32.U4(3).D8 36:2.M12 (holomorph of ternary Golay code) (A5 × J2):2 31+4:2.Sp4(3).2 (A6 × U3(3)).2 33+4:2.(S4 × S4) A9 × S3 (A7 × L2(7)):2 (order divisible by 49) (D10 × (A5 × A5).2).2 51+2:GL2(5) (contains Sylow 5-subgroups of Co1) 53:(4 × A5).2 (contains Sylow 5-subgroups of Co1) 72:(3 × 2.S4) (order divisible by 49) 52:2A5

Co1 contains non-abelian simple groups of some 35 isomorphism types, as subgroups or as quotients of subgroups.

Conway group

Maximal subgroups of Co2 There are 11 conjugacy classes of maximal subgroups. • • • • • • • • • • •

U6(2):2 210:M22:2 McL (fixing 2-2-3 triangle) 21+8:Sp6(2) HS:2 (can transpose type 3 vertices of conserved 2-3-3 triangle) (24 × 21+6).A8 U4(3):D8 24+10.(S5 × S3) M23 31+4.21+4.S5 51+2:4S4

Maximal subgroups of Co3 There are 14 conjugacy classes of maximal subgroups. Co3 has a doubly transitive permutation representation on 276 type 2-2-3 triangles containing a fixed type 3 point. • • • • • • • • • • • • • •

McL:2 - can transpose type 2 points of conserved 2-2-3 triangle HS - fixes 2-3-3 triangle U4(3).22 M23 35:(2 × M11) 2.Sp6(2) - centralizer of involution class 2A, which moves 240 of the 276 type 2-2-3 triangles U3(5):S3 31+4:4S6 24.A8 PSL(3,4):(2 × S3) 2 × M12 - centralizer of involution class 2B, which moves 264 of the 276 type 2-2-3 triangles [210.33] S3 × PSL(2,8):3 A4 × S5

A chain of product groups Co0 (as well as its quotient Co1) has 4 conjugacy clases of elements of order 3. One of these commutes with a double cover of the alternating group A9. In fact the normalizer of that 3-element has the form 2.A9 x S3. This maximal subgroup reveals interesting features not found in the Mathieu groups. It has a simple subgroup of order 504, containing an element of order 9. It was fruitful to investigate the normalizers of smaller subgroups of the form 2.An[2] . Several other maximal subgroups of Co0 are found in this way. Moreover, two sporadic groups appear in the resulting chain. There is a subgroup 2.A8 x S4, but it is not maximal in Co0. Next there is the subgroup (2.A7 x PSL2(7)):2, whose order is divisible by 49. This group and the rest of the chain are maximal in Co0. Next comes (2.A6 x SU3(3)):2. The unitary group SU3(3) (order 6048) possesses a graph of 36 vertices, in anticipation of the next subgroup. That subgroup is (2.A5 o 2.HJ):2. The aforementioned graph expands to the Hall-Janko graph, with 100 vertices. The Hall-Janko group HJ makes its appearance here. Next comes (2.A4 o 2.G2(4)):2. G2(4) is an exceptional group of Lie type. Its order is divisible by 13, fairly rare among subgroups of the Conway groups.

217

Conway group The chain ends with 6.Suz:2 (Suz=Suzuki group) , which, as mentioned above, respects a complex representaion of the Leech Lattice.

References [1] Atlas, both versions 2 & 3. [2] Robert A. Wilson, 'The Finite Simple Groups', Springer-Verlag (2009), p. 219 ff.

• Conway, John Horton (1968), "A perfect group of order 8,315,553,613,086,720,000 and the sporadic simple groups", Proceedings of the National Academy of Sciences of the United States of America 61: 398–400, doi:10.1073/pnas.61.2.398, MR0237634 • Richard Brauer and Chih-Han Shah, Theory of Finite Groups: A Symposium, W. A. Benjamin (1969) • Conway, J. H.: A group of order 8,315,553,613,086,720,000. Bull. London Math. Soc. 1 (1969), 79-88, the first-ever article on the group .0 • Conway, J. H.: Three lectures on exceptional groups, in Finite Simple Groups, M. B. Powell and G. Higman (editors), Academic Press, (1971), 215-247. Reprinted in J. H. Conway & N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer (1988), 267-298. • Thompson, Thomas M.: From Error Correcting Codes through Sphere Packings to Simple Groups, Carus Mathematical Monographs, Mathematical Association of America (1983). • Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; Wilson, R. A., Atlas of finite groups. Maximal subgroups and ordinary characters for simple groups. With computational assistance from J. G. Thackray. Eynsham: Oxford University Press (1985), ISBN 0-19-853199-0 • Griess, Robert L. Jr. (1998), Twelve sporadic groups, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, MR1707296, ISBN 978-3-540-62778-4 • Atlas of Finite Group Representations: Co1 (http://web.mat.bham.ac.uk/atlas/v2.0/spor/Co1/) version 2 • Atlas of Finite Group Representations: Co1 (http://brauer.maths.qmul.ac.uk/Atlas/v3/spor/Co1/) version 3 • Robert A. Wilson, 'The Finite Simple Groups', Springer-Verlag (2009).

218

Mathieu group

Mathieu group In the mathematical field of group theory, the Mathieu groups, named after the French mathematician Émile Léonard Mathieu, are five finite simple groups he discovered and reported in papers in 1861 and 1873; these were the first sporadic simple groups discovered. They are usually denoted by the symbols M11, M12, M22, M23, M24, and can be thought of respectively as permutation groups on sets of 11, 12, 22, 23 or 24 objects (or points). Sometimes the notation M7, M8, M9, M10, M19, M20 and M21 is used for related groups (which act on sets of 7, 8, 9, 10, 19, 20, and 21 points, respectively), namely the stabilizers of points in the larger groups. While these are not sporadic simple groups, they are important subgroups of the larger groups and can be used to construct the larger ones.[1] Conversely, John Conway has suggested that one can extend this sequence up by generalizing the fifteen puzzle, obtaining a subset of the symmetric group on 13 points denoted M13.[2] [3] M24, the largest of the groups, and which contains all the others, is contained within the symmetry group of the binary Golay code, which has practical uses. Moreover, the Mathieu groups are fascinating to many group theorists as mathematical anomalies.

History Simple groups are defined as having no nontrivial proper normal subgroups. Intuitively this means they cannot be broken down in terms of smaller groups. For many years group theorists struggled to classify the simple groups and had found all of them by about 1980. Simple groups belong to a number of infinite families except for 26 groups including the Mathieu groups, called sporadic simple groups. After the Mathieu groups no new sporadic groups were found until 1965, when the group J1 was discovered.

Multiply transitive groups Mathieu was interested in finding multiply transitive permutation groups, which will now be defined. For a natural number k, a permutation group G acting on n points is k-transitive if, given two sets of points a1, ... ak and b1, ... bk with the property that all the ai are distinct and all the bi are distinct, there is a group element g in G which maps ai to bi for each i between 1 and k. Such a group is called sharply k-transitive if the element g is unique (i.e. the action on k-tuples is regular, rather than just transitive). M24 is 5-transitive, and M12 is sharply 5-transitive, with the other Mathieu groups (simple or not) being the subgroups corresponding to stabilizers of m points, and accordingly of lower transitivity (M23 is 4-transitive, etc.). The only 4-transitive groups are the symmetric groups Sk for k at least 4, the alternating groups Ak for k at least 6, and the Mathieu groups M24, M23, M12 and M11. The full proof requires the classification of finite simple groups, but some special cases have been known for much longer. It is a classical result of Jordan that the symmetric and alternating groups (of degree k and k + 2 respectively), and M12 and M11 are the only sharply k-transitive permutation groups for k at least 4. Important examples of multiply transitive groups are the 2-transitive groups and the Zassenhaus groups. The Zassenhaus groups notably include the projective general linear group of a projective line over a finite field, PGL(2,Fq), which is sharply 3-transitive (see cross ratio) on elements.

219

Mathieu group

220

Order and transitivity table Group

Order

Order (product)

Factorised order

Transitivity

Simple

M24

244823040 3·16·20·21·22·23·24 210·33·5·7·11·23

5-transitive

simple

M23

10200960

3·16·20·21·22·23

27·32·5·7·11·23

4-transitive

simple

M22

443520

3·16·20·21·22

27·32·5·7·11

3-transitive

simple

M21

20160

3·16·20·21

26·32·5·7

2-transitive

simple

M20

960

3·16·20

26·3·5

1-transitive

not simple

M19

48

3·16

24·3

[4] 0-transitive

not simple

M12

95040

8·9·10·11·12

26·33·5·11

sharply 5-transitive simple

M11

7920

8·9·10·11

24·32·5·11

sharply 4-transitive simple

M10

720

8·9·10

24·32·5

sharply 3-transitive not simple

M9

72

8·9

23·32

sharply 2-transitive not simple

M8

8

8

23

sharply 1-transitive not simple

M7

1

1

1

sharply 0-transitive not simple

Constructions of the Mathieu groups The Mathieu groups can be constructed in various ways.

Permutation groups M12 has a simple subgroup of order 660, a maximal subgroup. That subgroup can be represented as a linear fractional group on the field F11 of 11 elements. With -1 written as a and infinity as b , two standard generators are (0123456789a) and (0b)(1a)(25)(37)(48)(69). A third generator giving M12 sends an element x of F11 to 4x2-3x7; as a permutation that is (26a7)(3945). The stabilizer of 4 points is a quaternion group. Likewise M24 has a maximal simple subgroup of order 6072 and this can be represented as a linear fractional group on the field F23. One generator adds 1 to each element (leaving the point N at infinity fixed), i. e. (0123456789ABCDEFGHIJKLM)(N), and the other is the order reversing permutation, (0N)(1M)(2B)(3F)(4H)(59)(6J)(7D)(8K)(AG)(CL)(EI). A third generator giving M24 sends an element x of F23 to 4x4-3x15; unexciting computation shows that as a permutation this is (2G968)(3CDI4)(7HABM)(EJLKF). These constructions were cited by Carmichael [5] ; Dixon and Mortimer ascribe the permutations to Mathieu. [6]

Mathieu group

Automorphism groups of Steiner systems There exists up to equivalence a unique S(5,8,24) Steiner system W24 (the Witt design). The group M24 is the automorphism group of this Steiner system; that is, the set of permutations which map every block to some other block. The subgroups M23 and M22 are defined to be the stabilizers of a single point and two points respectively. Similarly, there exists up to equivalence a unique S(5,6,12) Steiner system W12, and the group M12 is its automorphism group. The subgroup M11 is the stabilizer of a point. M24 from PSL(3,4) M24 can be built starting from PSL(3,4); this is one of the remarkable phenomena of mathematics. A good nest egg for M24 is PSL(3,4), the projective special linear group of 3-dimensional space over the finite field with 4 elements,[7] , also called M21 which acts on the projective plane over the field F4, an S(2,5,21) system called W21. Its 21 blocks are called lines. Any 2 lines intersect at one point. M21 has 168 simple subgroups of order 360 and 360 simple subgroups of order 168. In the larger projective general linear group PGL(3,4) both sets of subgroups form single conjugacy classes, but in M21 both sets split into 3 conjugacy classes. The subgroups respectively have orbits of 6, called hyperovals, and orbits of 7, called Fano subplanes. These sets allow creation of new blocks for larger Steiner systems. M21 is normal in PGL(3,4), of index 3. PGL(3,4) has an outer automorphism induced by transposing conjugate elements in F4 (the field automorphism). PGL(3,4) can therefore be extended to the group PΓL(3,4) of projective semilinear transformations, which is a split extension of M21 by the symmetric group S3. PΓL(3,4) turns out to have an embedding as a maximal subgroup of M24[8] . A hyperoval has no 3 points that are colinear. A Fano subplane likewise satisfies suitable uniqueness conditions . To W21 append 3 new points and let the automorphisms in PΓL(3,4) but not in M21 permute these new points. An S(3,6,22) system W22 is formed by appending just one new point to each of the 21 lines and new blocks are 56 hyperovals conjugate under M21. An S(5,8,24) system would have 759 blocks, or octads. Append all 3 new points to each line of W21, a different new point to the Fano subplanes in each of the sets of 120, and append appropriate pairs of new points to all the hyperovals. That accounts for all but 210 of the octads. Those remaining octads are subsets of W21 and are symmetric differences of pairs of lines. There are many possible ways to expand the group PΓL(3,4) to M24. W12 W12 can be constructed from the affine geometry on the vector space F3xF3, an S(2,3,9) system. An alternative construction of W12 is the 'Kitten' of R.T. Curtis.[9] Computer programs There have been notable computer programs written to generate Steiner systems. An introduction to a construction of W24 via the Miracle Octad Generator of R. T. Curtis and Conway's analog for W12, the miniMOG, can be found in the book by Conway and Sloane.

Automorphism group of the Golay code The group M24 also is the permutation automorphism group of the binary Golay code W, i.e., the group of permutations of coordinates mapping W to itself. Codewords correspond in a natural way to subsets of a set of 24 objects. Those subsets corresponding to codewords with 8 or 12 coordinates equal to 1 are called octads or dodecads respectively. The octads are the blocks of an S(5,8,24) Steiner system and the binary Golay code is the vector space over field F2 spanned by the octads of the Steiner system. The full automorphism group of the binary Golay code has order 212×|M24|, since there are |M24| permutations and 212 sign changes. These can be visualised by

221

Mathieu group

222

permuting and reflecting the coordinates on the vertices of a 24-dimensional cube. The simple subgroups M23, M22, M12, and M11 can be defined as subgroups of M24, stabilizers respectively of a single coordinate, an ordered pair of coordinates, a dodecad, and a dodecad together with a single coordinate. M12 has index 2 in its automorphism group. As a subgroup of M24, M12 acts on the second dodecad as an outer automorphic image of its action on the first dodecad. M11 is a subgroup of M23 but not of M22. This representation of M11 has orbits of 11 and 12. The automorphism group of M12 is a maximal subgroup of M24 of index 1288. There is a very natural connection between the Mathieu groups and the larger Conway groups, because the binary Golay code and the Leech lattice both lie in spaces of dimension 24. The Conway groups in turn are found in the Monster group. Robert Griess refers to the 20 sporadic groups found in the Monster as the Happy Family, and to the Mathieu groups as the first generation.

Dessins d'enfants The Mathieu groups can be constructed via dessins d'enfants, with the dessin associated to M12 suggestively called "Monsieur Mathieu".[10]

Polyhedral symmetries M24 can be constructed starting from the symmetries of the Klein quartic (the symmetries of a tessellation of the genus three surface), which is PSL(2,7), which can be augmented by an additional permutation. This permutation can be described by starting with the tiling of the Klein quartic by 20 triangles (with 24 vertices – the 24 points on which the group acts), then forming squares of out some of the 2 triangles, and octagons out of 6 triangles, with the added permutation being "interchange the two endpoints of the lines bisecting the squares and octagons". This can be visualized by coloring the triangles [11] – the corresponding tiling is topologically but not geometrically the t0,1{4, 3, 3} tiling, and can be (polyhedrally) immersed in Euclidean 3-space as the small cubicuboctahedron (which also has 24 vertices).[12]

M24 can be constructed from symmetries of the Klein quartic, augmented by a (non-geometric) symmetry of its immersion as the small cubicuboctahedron.

Properties The Mathieu groups have fascinating properties; these groups happen because of a confluence of several anomalies of group theory. For example, M12 contains a copy of the exceptional outer automorphism of S6. M12 contains a subgroup isomorphic to S6 acting differently on 2 sets of 6. In turn M12 has an outer automorphism of index 2 and, as a subgroup of M24, acts differently on 2 sets of 12. Note also that M10 is a non-split extension of the form A6.2 (an extension of the group of order 2 by A6), and accordingly A6 may be denoted M10′ as it is an index 2 subgroup of M10. The linear group GL(4,2) has an exceptional isomorphism to the alternating group A8; this isomorphism is important to the structure of M24. The pointwise stabilizer O of an octad is an abelian group of order 16, exponent 2, each of whose involutions moves all 16 points outside the octad. The stabilizer of the octad is a split extension of O by A8[13] . There are 759 (= 3·11·23) octads. Hence the order of M24 is 759*16*20160.

Mathieu group

Matrix representations in GL(11,2) The binary Golay code is a vector space of dimension 12 over F2. The fixed points under M24 form a subspace of 2 vectors, those with coordinates all 0 or all 1. The quotient space, of dimension 11, order 211, can be constructed as a set of partitions of 24 bits into pairs of Golay codewords. It is intriguing that the number of non-zero vectors, 211-1 = 2047, is the smallest Mersenne number with prime exponent that is not prime, equal to 23*89. Then |M24| divides |GL(11,2)| = 255*36*52*73*11*17*23*73*89. M23 also requires dimension 11. The groups M22, M12, and M11 are represented in GL(10,2).

Sextet subgroup of M24 Consider a tetrad, any set of 4 points in the Steiner system W24. An octad is determined by choice of a fifth point from the remaining 20. There are 5 octads possible. Hence any tetrad determines a partition into 6 tetrads, called a sextet, whose stabilizer in M24 is called a sextet group. The total number of tetrads is 24*23*22*21/4! = 23*22*21. Dividing that by 6 gives the number of sextets, 23*11*7 = 1771. Furthermore, a sextet group is a subgroup of a wreath product of order 6!*(4!)6, whose only prime divisors are 2, 3, and 5. Now we know the prime divisors of |M24|. Further analysis would determine the order of the sextet group and hence |M24|. It is convenient to arrange the 24 points into a 6-by-4 array: AEIMQU BFJNRV CGKOSW DHLPTX Moreover, it is convenient to use the elements of the field F4 to number the rows: 0, 1, u, u2. The sextet group has a normal abelian subgroup H of order 64, isomorphic to the hexacode, a vector space of length 6 and dimension 3 over F4. A non-zero element in H does double transpositions within 4 or 6 of the columns. Its action can be thought of as addition of vector co-ordinates to row numbers. The sextet group is a split extension of H by a group 3.S6 (a stem extension). Here is an instance within the Mathieu groups where a simple group (A6) is a subquotient, not a subgroup. 3.S6 is the normalizer in M24 of the subgroup generated by r=(BCD)(FGH)(JKL)(NOP)(RST)(VWX), which can be thought of as a multiplication of row numbers by u2. The subgroup 3.A6 is the centralizer of . Generators of 3.A6 are: (AEI)(BFJ)(CGK)(DHL)(RTS)(VWX) (rotating first 3 columns) (AQ)(BS)(CT)(DR)(EU)(FX)(GV)(HW) (AUEIQ)(BXGKT)(CVHLR)(DWFJS) (product of preceding two) (FGH)(JLK)(MQU)(NRV)(OSW)(PTX) (rotating last 3 columns) An odd permutation of columns, say (CD)(GH)(KL)(OP)(QU)(RV)(SX)(TW), then generates 3.S6. The group 3.A6 is isomorphic to a subgroup of SL(3,4) whose image in PSL(3,4) has been noted above as the hyperoval group. The applet Moggie [14] has a function that displays sextets in color.

223

Mathieu group

Subgroup structure M24 contains non-abelian simple subgroups of 13 isomorphism types: five classes of A5, four classes of PSL(3,2), two classes of A6, two classes of PSL(2,11), one class each of A7, PSL(2,23), M11, PSL(3,4), A8, M12, M22, M23, and M24.

Maximal subgroups of M24 Robert T. Curtis completed the search for maximal subgroups of M24 in (Curtis 1977), which had previously been mistakenly claimed in (Choi 1972b).[15] The list is as follows:[8] • • • •

M23, order 10200960 M22:2, order 887040, orbits of 2 and 22 24:A8, order 322560, orbits of 8 and 16: octad group M12:2, order 190080, transitive and imprimitive: dodecad group 6

Copy of M12 acting differently on 2 sets of 12, reflecting outer automorphism of M12 • 2 :(3.S6), order 138240: sextet group (vide supra) • PSL(3,4):S3, order 120960, orbits of 3 and 21

• 26:(PSL(3,2) x S3), order 64512, transitive and imprimitive: trio group Stabilizer of partition into 3 octads • PSL(2,23), order 6072: doubly transitive • Octern group, order 168, simple, transitive and imprimitive, 8 blocks of 3 Last maximal subgroup of M24 to be found. This group's 7-elements fall into 2 conjugacy classes of 24.

Maximal subgroups of M23 • M22, order 443520 • PSL(3,4):2, order 40320, orbits of 21 and 2 • 24:A7, order 40320, orbits of 7 and 16 Stabilizer of W23 block • A8, order 20160, orbits of 8 and 15 • M11, order 7920, orbits of 11 and 12 • (24:A5):S3 or M20:S3, order 5760, orbits of 3 and 20 (5 blocks of 4) One-point stabilizer of the sextet group • 23:11, order 253, simply transitive

224

Mathieu group

Maximal subgroups of M22 There are no proper subgroups transitive on all 22 points. • PSL(3,4) or M21, order 20160: one-point stabilizer • 24:A6, order 5760, orbits of 6 and 16 Stabilizer of W22 block • A7, order 2520, orbits of 7 and 15 • A7, orbits of 7 and 15 • 24:S5, order 1920, orbits of 2 and 20 (5 blocks of 4) A 2-point stabilizer in the sextet group • 23:PSL(3,2), order 1344, orbits of 8 and 14 • M10, order 720, orbits of 10 and 12 (2 blocks of 6) A one-point stabilizer of M11 (point in orbit of 11) A non-split extension of form A6.2 • PSL(2,11), order 660, orbits of 11 and 11 Another one-point stabilizer of M11 (point in orbit of 12)

Maximal subgroups of M21 There are no proper subgroups transitive on all 21 points. • 24:A5 or M20, order 960: one-point stabilizer Imprimitive on 5 blocks of 4 • • • • • • • •

4

2 :A5, transpose of M20, orbits of 5 and 16 A6, order 360, orbits of 6 and 15: hyperoval group A6, orbits of 6 and 15 A6, orbits of 6 and 15 PSL(3,2), order 168, orbits of 7 and 14: Fano subplane group PSL(3,2), orbits of 7 and 14 PSL(3,2), orbits of 7 and 14 32:Q or M9, order 72, orbits of 9 and 12

Maximal subgroups of M12 There are 11 conjugacy classes of maximal subgroups, 6 occurring in automorphic pairs. • M11, order 7920, degree 11 • M11, degree 12 Outer automorphic image of preceding type • S6:2, order 1440, imprimitive and transitive, 2 blocks of 6 Example of the exceptional outer automorphism of S6 • M10.2, order 1440, orbits of 2 and 10 • PSL(2,11), order 660, doubly transitive on the 12 points • 32:(2.S4), order 432, orbits of 3 and 9 Isomorphic to the affine group on the space C3 x C3. 2

• 3 :(2.S4), imprimitive on 4 sets of 3 • S5 x 2, order 240, doubly imprimitive, 6 by 2

225

Mathieu group

226

Centralizer of a sextuple transposition • Q:S4, order 192, orbits of 4 and 8. Centralizer of a quadruple transposition 2

• 4 :(2 x S3), order 192, imprimitive on 3 sets of 4 • A4 x S3, order 72, doubly imprimitive, 4 by 3

Maximal subgroups of M11 There are 5 conjugacy classes of maximal subgroups. • • • •

M10, order 720, one-point stabilizer in representation of degree 11 PSL(2,11), order 660, one-point stabilizer in representation of degree 12 M9:2, order 144, stabilizer of a 9 and 2 partition. S5, order 120, orbits of 5 and 6 Stabilizer of block in the S(4,5,11) Steiner system

• Q:S3, order 48, orbits of 8 and 3 Centralizer of a quadruple transposition Isomorphic to GL(2,3).

Number of elements of each order The maximum order of any element in M11 is 11. The conjugacy class orders and sizes are found in the ATLAS.[16] Order

No. elements

Conjugacy

1=1

1=1

1 class

2=2

165 = 3 · 5 · 11

1 class

3=3

440 = 23 · 5 · 11

1 class

4 = 22

990 = 2 · 32 · 5 · 11

1 class

5=5

1584 = 24 · 32 · 11

1 class

6 = 2 · 3 1320 = 23 · 3 · 5 · 11

1 class

8 = 23

1980 = 22 · 32 · 5 · 11 2 classes (power equivalent)

11 = 11

1440 = 25 · 32 · 5

2 classes (power equivalent)

The maximum order of any element in M12 is 11. The conjugacy class orders and sizes are found in the ATLAS[17].

Mathieu group

227

Order

No. elements

Conjugacy

1=1

1=1

1 class

2=2

891 = 34 · 11

2 classes (not power equivalent)

3=3

4400 = 24 · 52 · 11

2 classes (not power equivalent)

4 = 22

5940 = 22 · 33 · 5 · 11

2 classes (not power equivalent)

5=5

9504 = 25 · 33 · 11

1 class

6=2·3

23760 = 24 · 33 · 5 · 11 2 classes (not power equivalent)

8 = 23

23760 = 24 · 33 · 5 · 11 2 classes (not power equivalent)

10 = 2 · 5 9504 = 25 · 33 · 11

1 class

11 = 11

2 classes (power equivalent)

17280 = 27 · 33 · 5

The maximum order of any element in M22 is 11. Order

No. elements

Conjugacy

1=1

1=1

1 class

2=2

1155 = 3 · 5 · 7 · 11

1 class

3=3

12320 = 25 · 5 · 7 · 11

1 class

4 = 22

13860 = 22 · 32 · 5 · 7 · 11 1 class 27720 = 23 · 32 · 5 · 7 · 11 1 class

5=5

88704 = 27 · 32 · 7 · 11

1 class

6 = 2 · 3 36960 = 25 · 3 · 5 · 7 · 11

1 class

7=7

126720 = 28 · 32 · 5 · 11

2 classes, power equivalent

8 = 23

55440 = 24 · 32 · 5 · 7 · 11 1 class

11 = 11

80640 = 28 · 32 · 5 · 7

2 classes, power equivalent

The maximum order of any element in M23 is 23. Order

No. elements

Conjugacy

1=1

1=1

1 class

2=2

3795 = 3 · 5 · 11 · 23

1 class

3=3

56672 = 25 · 7 · 11 · 23

1 class

4 = 22

318780 = 22 · 32 · 5 · 7 · 11 · 23

1 class

5=5

680064 = 27 · 3 · 7 · 11 · 23

1 class

6=2·3

850080 = 25 · 3 · 5 · 7 · 11 · 23

1 class

7=7

1457280 = 27 · 32 · 5 · 11 · 23

2 classes, power equivalent

8 = 23

1275120 = 24 · 32 · 5 · 7 · 11 · 23 1 class

Mathieu group

228 11 = 11

1854720 = 28 · 32 · 5 · 7 · 23

2 classes, power equivalent

14 = 2 · 7 1457280 = 27 · 32 · 5 · 11 · 23

2 classes, power equivalent

15 = 3 · 5 1360128 = 28 · 3 · 7 · 11 · 23

2 classes, power equivalent

23 = 23

2 classes, power equivalent

887040 = 28 · 32 · 5 · 7 · 11

The maximum order of any element in M24 is 23. There are 26 conjugacy classes. Order

No. elements

Cycle structure and conjugacy

1=1

1

1 class

2=2

11385 = 32 · 5 · 11 · 23

28, 1 class

31878 = 2 · 32 · 7 · 11 · 23

212, 1 class

226688 = 27 · 7 · 11 · 23

36, 1 class

485760 = 27 · 3 · 5 · 11 · 23

38, 1 class

637560 = 23 · 32 · 5 · 7 · 11 · 23

2444, 1 class

1912680 = 23 · 33 · 5 · 7 · 11 · 23

2244, 1 class

2550240 = 25 · 32 · 5 · 7 · 11 · 23

46, 1 class

5=5

4080384 = 28 · 33 · 7 · 11 · 23

54, 1 class

6=2·3

10200960 = 27 · 32 · 5 · 7 · 11 · 23 223262, 1 class

3=3

4 = 22

10200960 = 27 · 32 · 5 · 7 · 11 · 23 2444, 1 class 7=7

11658240 = 210 · 32 · 5 · 11 · 23

8 = 23

15301440 = 26 · 33 · 5 · 7 · 11 · 23 2·4·82, 1 class

10 = 2 · 5

12241152 = 28 · 33 · 7 · 11 · 23

22102, 1 class

11 = 11

22256640 = 210 · 33 · 5 · 7 · 23

112, 1 class

73, 2 power equivalent classes

12 = 22 · 3 20401920 = 28 · 32 · 5 · 7 · 11 · 23 2 ·4·6·12, 1 class 20401920 = 28 · 32 · 5 · 7 · 11 · 23 122, 1 class 14 = 2 · 7

34974720 = 210 · 33 · 5 · 11 · 23

2·7·14, 2 power equivalent classes

15 = 3 · 5

32643072 = 211 · 32 · 7 · 11 · 23

3·5·15, 2 power equivalent classes

21 = 3 · 7

23316480 = 211 · 32 · 5 · 11 · 23

3·21, 2 power equivalent classes

23 = 23

21288960 = 211 · 33 · 5 · 7 · 11

23, 2 power equivalent classes

Mathieu group

229

Notes [1] M7 is the trivial group, while M19 does not act transitively on 19 points and 19 does not divide its order, so this sequence cannot be extended further down. [2] John H. Conway, "Graphs and Groups and M13", Notes from New York Graph Theory Day XIV (1987), pp. 18–29. [3] Conway, John Horton; Elkies, Noam D.; Martin, Jeremy L. (2006), "The Mathieu group M12 and its pseudogroup extension M13" (http:/ / nrs. harvard. edu/ urn-3:HUL. InstRepos:2794826), Experimental Mathematics 15 (2): 223–236, MR2253008, ISSN 1058-6458, [4] M19 acts non-trivially but intransitively on 19 points, and has order 3·16; note that

In fact, it has 2 orbits: one of order 16,

one of order 3 (the Sylow 2-subgroup acts regularly on 16 points, fixing the other 3, while the Sylow 3-subgroup permutes the 3 points, fixing the order 16 orbit). See (Choi 1972a, p. 4) for details. [5] Carmichael (1937): pp.151, 164, 263. [6] Dixon and Mortimer (1996): p. 209. [7] (Dixon & Mortimer 1996, pp. 192–205) [8] (Griess 1998, p. 55 (http:/ / books. google. com/ books?id=Ue2pJaegL50C& pg=PA55)) [9] (Curtis 1984) [10] le Bruyn, Lieven (01 March 2007), Monsieur Mathieu (http:/ / www. neverendingbooks. org/ index. php/ monsieur-mathieu. html), . [11] http:/ / homepages. wmich. edu/ ~drichter/ images/ mathieu/ hypercolors. jpg [12] (Richter) [13] Thomas Thompson (1983), pp. 197-208. [14] http:/ / nickerson. org. uk/ groups/ moggie/ [15] (Griess 1998, p. 54 (http:/ / books. google. com/ books?id=Ue2pJaegL50C& pg=PA54)) [16] ATLAS: Mathieu group M11 (http:/ / brauer. maths. qmul. ac. uk/ Atlas/ spor/ M11/ ) [17] http:/ / brauer. maths. qmul. ac. uk/ Atlas/ spor/ M12/

References • Mathieu E., Mémoire sur l'étude des fonctions de plusieurs quantités, sur la manière de les former et sur les substitutions qui les laissent invariables J. Math. Pures Appl. (Liouville) (2) VI, 1861, pp. 241-323. • Mathieu E., Sur la fonction cinq fois transitive de 24 quantités, Liouville Journ., (2) XVIII., 1873, pp. 25-47. • Carmichael, Robert D. Groups of Finite Order, Dover (1937, reprint 1956). • Conway, J.H.; Sloane N.J.A. Sphere Packings, Lattices and Groups: v. 290 (Grundlehren Der Mathematischen Wissenschaften.) Springer Verlag. ISBN 0-387-98585-9 • Choi, C. (May 1972a), "On Subgroups of M24. I: Stabilizers of Subsets" (http://jstor.org/stable/1996123), Transactions of the American Mathematical Society (American Mathematical Society) 167: 1–27, doi:10.2307/1996123, JSTOR 1996123 • Choi, C. (May 1972b). "On Subgroups of M24. II: the Maximal Subgroups of M24" (http://jstor.org/stable/ 1996124). Transactions of the American Mathematical Society (American Mathematical Society) 167: 29–47. doi:10.2307/1996124. JSTOR 1996124. • Curtis, R. T. A new combinatorial approach to M24. Math. Proc. Camb. Phil. Soc. 79 (1976) 25-42. • Curtis, R. T. The maximal subgroups of M24. Math. Proc. Camb. Phil. Soc. 81 (1977) 185-192. • Thompson, Thomas M.: From Error Correcting Codes through Sphere Packings to Simple Groups, Carus Mathematical Monographs, Mathematical Association of America, 1983. • Curtis, R. T. The Steiner System S(5,6,12), the Mathieu Group M12 and the 'Kitten', Computational Group Theory, Academic Press, London, 1984 • Cuypers, Hans, The Mathieu groups and their geometries (http://www.win.tue.nl/~hansc/mathieu.pdf) • Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; Wilson, R. A. (1985). Atlas of finite groups. Maximal subgroups and ordinary characters for simple groups. With computational assistance from J. G. Thackray. Eynsham: Oxford University Press. ISBN 0-19-853199-0 • ATLAS: Mathieu group M10 (http://brauer.maths.qmul.ac.uk/Atlas/v3/group/M10/) • ATLAS: Mathieu group M11 (http://brauer.maths.qmul.ac.uk/Atlas/v3/group/M11/) • ATLAS: Mathieu group M12 (http://brauer.maths.qmul.ac.uk/Atlas/v3/group/M12/) • ATLAS: Mathieu group M20 (http://brauer.maths.qmul.ac.uk/Atlas/v3/group/M20/)

Mathieu group

• • • •

• ATLAS: Mathieu group M21 (http://brauer.maths.qmul.ac.uk/Atlas/v3/group/M21/) • ATLAS: Mathieu group M22 (http://brauer.maths.qmul.ac.uk/Atlas/v3/group/M22/) • ATLAS: Mathieu group M23 (http://brauer.maths.qmul.ac.uk/Atlas/v3/group/M23/) • ATLAS: Mathieu group M24 (http://brauer.maths.qmul.ac.uk/Atlas/v3/group/M24/) Dixon, John D.; Mortimer, Brian (1996), Permutation Groups, Springer-Verlag Griess, Robert L.: Twelve Sporadic Groups, Springer-Verlag, 1998. Ronan M. "Symmetry and the Monster", Oxford University Press (2006) ISBN 0-19-280722-6 (an introduction for the lay reader, describing the Mathieu groups in a historical context) Richter, David A., How to Make the Mathieu Group M24 (http://homepages.wmich.edu/~drichter/mathieu. htm), retrieved 2010-04-15

External links • Moggie (http://nickerson.org.uk/groups/moggie/) Java applet for studying the Curtis MOG construction • Scientific American (http://www.sciam.com/article.cfm?id=puzzles-simple-groups-at-play) A set of puzzles based on the mathematics of the Mathieu groups • Sporadic M12 (http://itunes.apple.com/us/app/sporadic-m12/id322438247) An iPhone app that implements puzzles based on M12, presented as one "spin" permutation and a selectable "swap" permutation • Octad of the week (http://igor.gold.ac.uk/~mas01rwb/octad.html)

Sporadic groups In the mathematical field of group theory, a sporadic group is one of the 26 exceptional groups in the classification of finite simple groups. A simple group is a group G that does not have any normal subgroups except for the subgroup consisting only of the identity element, and G itself. The classification theorem states that the list of finite simple groups consists of 18 countably infinite families, plus 26 exceptions that do not follow such a systematic pattern. These are the sporadic groups. They are also known as the sporadic simple groups, or the sporadic finite groups. Sometimes the Tits group is regarded as a sporadic group (because it is not strictly a group of Lie type), in which case there are 27 sporadic groups. The Monster group is the largest of the sporadic groups and contains all but six of the other sporadic groups as subgroups or subquotients.

230

Sporadic groups

231

Names of the sporadic groups Five of the sporadic groups were discovered by Mathieu in the 1860s and the other 21 were found between 1965 and 1975. Several of these groups were predicted to exist before they were constructed. Most of the groups are named after the mathematician(s) who first predicted their existence. The full list is: • Mathieu groups M11, M12, M22, M23, M24 • Janko groups J1, J2 or HJ, J3 or HJM, J4 • Conway groups Co1 or F2−, Co2, Co3 • Fischer groups Fi22, Fi23, Fi24′ or F3+ • Higman–Sims group HS • McLaughlin group McL • Held group He or F7+ or F7 • Rudvalis group Ru • Suzuki sporadic group Suz or F3− • O'Nan group O'N • Harada–Norton group HN or F5+ or F5 • Lyons group Ly • Thompson group Th or F3|3 or F3

Sporadic Finite Groups Showing (Sporadic) Subgroups

• Baby Monster group B or F2+ or F2 • Fischer–Griess Monster group M or F1 Matrix representations over finite fields for all the sporadic groups have been computed. The earliest use of the term "sporadic group" may be Burnside (1911, p. 504, note N) where he comments about the Mathieu groups: "These apparently sporadic simple groups would probably repay a closer examination than they have yet received". Diagram is based on diagram given in Ronan (2006). The sporadic groups also have a lot of subgroups which are not sporadic but these are not shown on the diagram because they are too numerous.

Organization Of the 26 sporadic groups, 20 can be seen inside the Monster group as subgroups or quotients of subgroups. The six exceptions are J1, J3, J4, O'N, Ru and Ly. These six groups are sometimes known as the pariahs. The remaining twenty groups have been called the Happy Family by Robert Griess, and can be organized into three generations.

First generation: the Mathieu groups The Mathieu groups Mn (for n = 11, 12, 22, 23 and 24) are multiply transitive permutation groups on n points. They are all subgroups of M24, which is a permutation group on 24 points.

Sporadic groups

232

Second generation: the Leech lattice The second generation are all subquotients of the automorphism group of a lattice in 24 dimensions called the Leech lattice: • • • • • • •

Co1 is the quotient of the automorphism group by its center {±1} Co2 is the stabilizer of a type 2 (i.e., length 2) vector Co3 is the stabilizer of a type 3 (i.e., length √6) vector Suz is the group of automorphisms preserving a complex structure (modulo its center) McL is the stabilizer of a type 2-2-3 triangle HS is the stabilizer of a type 2-3-3 triangle J2 is the group of automorphisms preserving a quaternionic structure (modulo its center).

Third generation: other subgroups of the Monster The third generation consists of subgroups which are closely related to the Monster group M: • B or F2 has a double cover which is the centralizer of an element of order 2 in M • Fi24′ has a triple cover which is the centralizer of an element of order 3 in M (in conjugacy class "3A") • Fi23 is a subgroup of Fi24′ • Fi22 has a double cover which is a subgroup of Fi23 • The product of Th = F3 and a group of order 3 is the centralizer of an element of order 3 in M (in conjugacy class "3C") • The product of HN = F5 and a group of order 5 is the centralizer of an element of order 5 in M • The product of He = F7 and a group of order 7 is the centralizer of an element of order 7 in M. • Finally, the Monster group itself is considered to be in this generation. (This series continues further: the product of M12 and a group of order 11 is the centralizer of an element of order 11 in M.) The Tits group also belongs in this generation: there is a subgroup S4 ×2F4(2)′ normalising a 2C2 subgroup of B, giving rise to a subgroup 2·S4 ×2F4(2)′ normalising a certain Q8 subgroup of the Monster. 2F4(2)′ is also a subgroup of the Fischer groups Fi22, Fi23 and Fi24′, and of the Baby Monster B. 2F4(2)′ is also a subgroup of the (pariah) Rudvalis group Ru, and has no involvements in sporadic simple groups except the containments we have already mentioned.

Table of the sporadic group orders Group

Order (sequence A001228

[1]

in OEIS)

1SF

F1 or M

808017424794512875886459904961710757005754368000000000 ≈ 8×1053

F2 or B

4154781481226426191177580544000000 ≈ 4×1033

Fi24' or F3+

1255205709190661721292800 ≈ 1×1024

Fi23

4089470473293004800 ≈ 4×1018

Fi22

64561751654400 ≈ 6×1013

Factorized order 246 · 320 · 59 · 76 · 112 · 133 · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71 241 · 313 · 56 · 72 · 11 · 13 · 17 · 19 · 23 · 31 · 47

221 · 316 · 52 · 73 · 11 · 13 · 17 · 23 · 29

218 · 313 · 52 · 7 · 11 · 13 · 17 · 23

217 · 39 · 52 · 7 · 11 · 13

Sporadic groups

233

F3 or Th

90745943887872000 ≈ 9×1016

215 · 310 · 53 · 72 · 13 · 19 · 31

Ly

51765179004000000 ≈ 5×1016

28 · 37 · 56 · 7 · 11 · 31 · 37 · 67

F5 or HN

273030912000000 ≈ 3×1014

214 · 36 · 56 · 7 · 11 · 19

Co1

4157776806543360000 ≈ 4×1018

Co2

42305421312000 ≈ 4×1013

218 · 36 · 53 · 7 · 11 · 23

Co3

495766656000 ≈ 5×1011

210 · 37 · 53 · 7 · 11 · 23

O'N

460815505920 ≈ 5×1011

29 · 34 · 5 · 73 · 11 · 19 · 31

Suz

448345497600 ≈ 4×1011

213 · 37 · 52 · 7 · 11 · 13

Ru

145926144000 ≈ 1×1011

214 · 33 · 53 · 7 · 13 · 29

He McL HS J4

J3 or HJM

221 · 39 · 54 · 72 · 11 · 13 · 23

4030387200 ≈ 4×109 210 · 33 · 52 · 73 · 17 898128000 ≈ 9×108 27 · 36 · 53 · 7 · 11 44352000 ≈ 4×107 29 · 32 · 53 · 7 · 11 86775571046077562880 ≈ 9×1019

221 · 33 · 5 · 7 · 113 · 23 · 29 · 31 · 37 · 43

50232960 ≈ 5×107 27 · 35 · 5 · 17 · 19

J2 or HJ

604800 ≈ 6×105 27 · 33 · 52 · 7

J1

175560 ≈ 2×105 23 · 3 · 5 · 7 · 11 · 19

M24

244823040 ≈ 2×108 210 · 33 · 5 · 7 · 11 · 23

M23

10200960 ≈ 1×107 27 · 32 · 5 · 7 · 11 · 23

M22

443520 ≈ 4×105 27 · 32 · 5 · 7 · 11

M12

95040 ≈ 1×105 26 · 33 · 5 · 11

M11

7920 ≈ 8×103 24 · 32 · 5 · 11

Sporadic groups

References • Burnside, William (1911), Theory of groups of finite order, pp. 504 (note N), ISBN 0486495752 (2004 reprinting) • Conway, J. H.: A perfect group of order 8,315,553,613,086,720,000 and the sporadic simple groups, Proc. Nat. Acad. Sci. U.S.A. 61 (1968), 398–400. • Conway, J. H.: Curtis, R. T.; Norton, S. P.; Parker, R. A.; Wilson, R. A., Atlas of finite groups. Maximal subgroups and ordinary characters for simple groups. With computational assistance from J. G. Thackray. Eynsham: Oxford University Press, 1985, ISBN 0-19-853199-0 • Daniel Gorenstein, Richard Lyons, Ronald Solomon The Classification of the Finite Simple Groups (volume 1) [2] , AMS, 1994 (volume 2) [3], AMS. • Griess, Robert L.: "Twelve Sporadic Groups", Springer-Verlag, 1998. • Ronan, Mark (2006), Symmetry and the Monster, Oxford, ISBN 978-0-19-280722-9

External links • Weisstein, Eric W., "Sporadic Group [4]" from MathWorld. • Atlas of Finite Group Representations: Sporadic groups [5]

References [1] [2] [3] [4] [5]

http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa001228 http:/ / www. ams. org/ online_bks/ surv401/ http:/ / www. ams. org/ online_bks/ surv402/ http:/ / mathworld. wolfram. com/ SporadicGroup. html http:/ / brauer. maths. qmul. ac. uk/ Atlas/ v3/ spor/

Janko group J1 In mathematics, the smallest Janko group, J1, of order 175560, was first described by Zvonimir Janko (1965), in a paper which described the first new sporadic simple group to be discovered in over a century and which launched the modern theory of sporadic simple groups.

Properties J1 can be characterized abstractly as the unique simple group with abelian 2-Sylow subgroups and with an involution whose centralizer is isomorphic to the direct product of the group of order two and the alternating group A5 of order 60, which is to say, the rotational icosahedral group. That was Janko's original conception of the group. In fact Janko and Thompson were investigating groups similar to the Ree groups 2G2(32n+1), and showed that if a simple group G has abelian Sylow 2-subgroups and a centralizer of an involution of the form Z/2Z×PSL2(q) for q a prime power at least 3, then either q is a power of 3 and G has the same order as a Ree group (it was later shown that G must be a Ree group in this case) or q is 4 or 5. Note that PSL2(4)=PSL2(5)=A5. This last exceptional case led to the Janko group J1. J1 has no outer automorphisms and its Schur multiplier is trivial. J1 is the smallest of the 6 sporadic simple groups called the pariahs, because they are not found within the Monster group. J1 is contained in the O'Nan group as the subgroup of elements fixed by an outer automorphism of order 2.

234

Janko group J<sub>1

235

Construction Janko found a modular representation in terms of 7 × 7 orthogonal matrices in the field of eleven elements, with generators given by

and

Y has order 7 and Z has order 5. Janko (1966) credited W. A. Coppel for recognizing this representation as an embedding into Dickson's simple group G2(11) (which has a 7 dimensional representation over the field with 11 elements). There is also a pair of generators a, b such that a2=b3=(ab)7=(abab−1)19=1 J1 is thus a Hurwitz group, a finite homomorphic image of the (2,3,7) triangle group.

Maximal subgroups Janko (1966) enumerated all 7 conjugacy classes of maximal subgroups (see also the Atlas webpages cited below). Maximal simple subgroups of order 660 afford J1 a permutation representation of degree 266. He found that there are 2 conjugacy classes of subgroups isomorphic to the alternating group A5, both found in the simple subgroups of order 660. J1 has non-abelian simple proper subgroups of only 2 isomorphism types. Here is a complete list of the maximal subgroups. Structure Order Index

Description

PSL2(11)

660

266

Fixes point in smallest permutation representation

23.7.3

168

1045

Normalizer of Sylow 2-subgroup

2×A5

120

1463

Centralizer of involution

19.6

114

1540

Normalizer of Sylow 19-subgroup

11.10

110

1596

Normalizer of Sylow 11-subgroup

D6×D10

60

2926 Normalizer of Sylow 3-subgroup and Sylow 5-subgroup

7.6

42

4180

Normalizer of Sylow 7-subgroup

The notation A.B means a group with a normal subgroup A with quotient B, and D2n is the dihedral group of order 2n.

Janko group J<sub>1

236

Number of elements of each order The greatest order of any element of the group is 19. The conjugacy class orders and sizes are found in the ATLAS. Order

No. elements

Conjugacy

1=1

1=1

1 class

2=2

1463 = 7 · 11 · 19

1 class

3=3

5852 = 22 · 7 · 11 · 19

1 class

5=5

11704 = 23 · 7 · 11 · 19

2 classes, power equivalent

6=2·3

29260 = 22 · 5 · 7 · 11 · 19 1 class

7=7

25080 = 23 · 3 · 5 · 11 · 19 1 class

10 = 2 · 5 35112 = 23 · 3 · 7 · 11 · 19 2 classes, power equivalent 11 = 11

15960 = 23 · 3 · 5 · 7 · 19

15 = 3 · 5 23408 = 24 · 7 · 11 · 19 19 = 19

1 class 2 classes, power equivalent

27720 = 23 · 32 · 5 · 7 · 11 3 classes, power equivalent

References • Zvonimir Janko, A new finite simple group with abelian Sylow subgroups, Proc. Nat. Acad. Sci. USA 53 (1965) 657-658. • Zvonimir Janko, A new finite simple group with abelian Sylow subgroups and its characterization, Journal of Algebra 3: 147-186, (1966) doi:10.1016/0021-8693(66)90010-X • Zvonimir Janko and John G. Thompson, On a Class of Finite Simple Groups of Ree, Journal of Algebra, 4 (1966), 274-292. • Robert A. Wilson, Is J1 a subgroup of the monster?, Bull. London Math. Soc. 18, no. 4 (1986), 349-350. • Atlas of Finite Group Representations: J1 [1] version 2 • Atlas of Finite Group Representations: J1 [2] version 3

References [1] http:/ / web. mat. bham. ac. uk/ atlas/ v2. 0/ spor/ J1/ [2] http:/ / brauer. maths. qmul. ac. uk/ Atlas/ v3/ spor/ J1/

Janko group J2

Janko group J2 In mathematics, the Hall-Janko group HJ, is a finite simple sporadic group of order 604800. It is also called the second Janko group J2, or the Hall-Janko-Wales group, since it was predicted by Janko and constructed by Hall and Wales. It is a subgroup of index two of the group of automorphisms of the Hall-Janko graph, leading to a permutation representation of degree 100. It has a modular representation of dimension six over the field of four elements; if in characteristic two we have w2 + w + 1 = 0, then J2 is generated by the two matrices

and

These matrices satisfy the equations

J2 is thus a Hurwitz group, a finite homomorphic image of the (2,3,7) triangle group. The matrix representation given above constitutes an embedding into Dickson's group G2(4). There are two conjugacy classes of HJ in G2(4), and they are equivalent under the automorphism on the field F4. Their intersection (the "real" subgroup) is simple of order 6048. G2(4) is in turn isomorphic to a subgroup of the Conway group Co1. J2 is the only one of the 4 Janko groups that is a section of the Monster group; it is thus part of what Robert Griess calls the Happy Family. Since it is also found in the Conway group Co1, it is therefore part of the second generation of the Happy Family. Griess relates [p. 123] how Marshall Hall, as editor of The Journal of Algebra, received a very short paper entitled "A simple group of order 604801." Yes, 604801 is prime. J2 has 9 conjugacy classes of maximal subgroups. Some are here described in terms of action on the Hall-Janko graph. • U3(3) order 6048 - one-point stabilizer, with orbits of 36 and 63 Simple, containing 36 simple subgroups of order 168 and 63 involutions, all conjugate, each moving 80 points. A given involution is found in 12 168-subgroups, thus fixes them under conjugacy. Its centralizer has structure 4.S4, which contains 6 additional involutions. • 3.PGL(2,9) order 2160 - has a subquotient A6 • 21+4:A5 order 1920 - centralizer of involution moving 80 points • 22+4:(3 × S3) order 1152 • A4 × A5 order 720 Containing 22 × A5 (order 240), centralizer of 3 involutions each moving 100 points

237

Janko group J2

238

• A5 × D10 order 600 • PGL(2,7) order 336 • 52:D12 order 300 • A5 order 60 Janko predicted both J2 and J3 as simple groups having 21+4:A5 as a centralizer of an involution.

Number of elements of each order The maximum order of any element is 15. As permutations, elements act on the 100 vertices of the Hall-Janko graph. Order

No. elements

Cycle structure and conjugacy

1=1

1=1

1 class

2=2

315 = 32 · 5 · 7

240, 1 class

2520 = 23 · 32 · 5 · 7

250, 1 class

560 = 24 · 5 · 7

330, 1 class

16800 = 25 · 3 · 52 · 7

332, 1 class

4 = 22

6300 = 22 · 32 · 52 · 7

26420, 1 class

5=5

4032 = 26 · 32 · 7

520, 2 classes, power equivalent

24192 = 27 · 33 · 7

520, 2 classes, power equivalent

3=3

6=2·3

25200 = 24 · 32 · 52 · 7 2436612, 1 class 50400 = 25 · 32 · 52 · 7 22616, 1 class

7=7

86400 = 27 · 33 · 52

8 = 23

75600 = 24 · 33 · 52 · 7 2343810, 1 class

10 = 2 · 5

60480 = 26 · 33 · 5 · 7

714, 1 class

1010, 2 classes, power equivalent

120960 = 27 · 33 · 5 · 7 54108, 2 classes, power equivalent 12 = 22 · 3 50400 = 25 · 32 · 52 · 7 324262126, 1 class 15 = 3 · 5

80640 = 28 · 32 · 5 · 7

52156, 2 classes, power equivalent

References • Robert L. Griess, Jr., "Twelve Sporadic Groups", Springer-Verlag, 1998. • Marshall Hall, Jr. and David Wales, "The Simple Group of Order 604,800", Journal of Algebra, 9 (1968), 417-450. • Wales, David B., "The uniqueness of the simple group of order 604800 as a subgroup of SL(6,4)", Journal of Algebra 11 (1969), 455 - 460. • Wales, David B., "Generators of the Hall-Janko group as a subgroup of G2(4)", Journal of Algebra 13 (1969), 513–516, doi:10.1016/0021-8693(69)90113-6, MR0251133, ISSN 0021-8693 • Z. Janko, Some new finite simple groups of finite order, 1969 Symposia Mathematica (INDAM, Rome, 1967/68), Vol. 1 pp. 25-64 Academic Press, London MR0244371 • Atlas of Finite Group Representations: J2 [1]

Janko group J2

239

References [1] http:/ / web. mat. bham. ac. uk/ atlas/ v2. 0/ spor/ J2/

Janko group J3 In mathematics, the third Janko group J3, also known as the Higman-Janko-McKay group, is a finite simple sporadic group of order 50232960. Evidence for its existence was uncovered by Zvonimir Janko (1969), and it was shown to exist by Graham Higman and John McKay (1969). Janko predicted both J3 and J2 as simple groups having 21+4:A5 as a centralizer of an involution. J3 has an outer automorphism group of order 2 and a Schur multiplier of order 3, and its triple cover has a unitary 9 dimensional representation over the field with 4 elements. Weiss (1982) constructed it via an underlying geometry. and it has a modular representation of dimension eighteen over the finite field of nine elements. J3 is one of the 6 sporadic simple groups called the pariahs, because (Greiss 1982) showed that it is not found within the Monster group.

Presentations In A

terms

of

generators

presentation

for

a, J3

b,

c, in

and terms

d

its of

automorphism (different)

group

J3:2

generators

can a,

be b,

presented

as

c,

is

d

Maximal subgroups Finkelstein & Rudvalis (1974) showed that J3 has 9 conjugacy classes of maximal subgroups: • • • • • • • • •

PSL(2,16):2, order 8160 PSL(2,19), order 3420 PSL(2,19), conjugate to preceding class in J3:2 24:(3 × A5), order 2880 PSL(2,17), order 2448 (3 × A6):22, order 2160 - normalizer of subgroup of order 3 32+1+2:8, order 1944 - normalizer of Sylow 3-subgroup 21+4:A5, order 1920 - centralizer of involution 22+4:(3 × S3), order 1152

References • Finkelstein, L.; Rudvalis, A. (1974), "The maximal subgroups of Janko's simple group of order 50,232,960", Journal of Algebra 30: 122–143, doi:10.1016/0021-8693(74)90196-3, MR0354846, ISSN 0021-8693 • R. L. Griess, Jr., The Friendly Giant, Inventiones Mathematicae 69 (1982), 1-102. p. 93: proof that J3 is a pariah. • Higman, Graham; McKay, John (1969), "On Janko's simple group of order 50,232,960", Bull. London Math. Soc. 1: 89–94; correction p. 219, doi:10.1112/blms/1.1.89, MR0246955 • Z. Janko, Some new finite simple groups of finite order, 1969 Symposia Mathematica (INDAM, Rome, 1967/68), Vol. 1 pp. 25-64 Academic Press, London, and in The theory of finite groups (Editied by Brauer and Sah) p. 63-64, Benjamin, 1969.MR0244371 • Richard Weiss, "A Geometric Construction of Janko's Group J3", Math. Zeitung 179 pp 91-95 (1982)

Janko group J<sub>3

External links • Atlas of Finite Group Representations: J3 [1] version 2 • Atlas of Finite Group Representations: J3 [2] version 3

References [1] http:/ / web. mat. bham. ac. uk/ atlas/ v2. 0/ spor/ J3/ [2] http:/ / brauer. maths. qmul. ac. uk/ Atlas/ v3/ spor/ J3/

Janko group J4 In mathematics, the fourth Janko group J4 is the sporadic finite simple group of order 221 · 33 · 5 · 7 · 113 · 23 · 29 · 31 · 37 · 43 = 86775571046077562880 whose existence was suggested by Zvonimir Janko (1976). Its existence and uniqueness was shown by Simon Norton and others in 1980. Janko found it by studying groups with an involution centralizer of the form 21+12.3.(M22:2). It has a modular representation of dimension 112 over the finite field of two elements and is the stabilizer of a certain 4995 dimensional subspace of the exterior square, a fact which Norton used to construct it, and which is the easiest way to deal with it computationally. The Schur multiplier and the outer automorphism group are both trivial. Ivanov (2004) has given a proof of existence and uniqueness that does not rely on computer calculations. J4 is one of the 6 sporadic simple groups called the pariahs, because they are not found within the Monster group. The order of the monster group is not divisible by 37 or 43.

Presentation It has a presentation in terms of three generators a, b, and c as

Maximal subgroups Kleidman & Wilson (1988) showed that J4 has 13 conjugacy classes of maximal subgroups.

• 211:M24 - containing Sylow 2-subgroups and Sylow 3-subgroups; also containing 211:(M22:2), centralizer of involution of class 2B • 21+12.3.(M22:2) - centralizer of involution of class 2A - containing Sylow 2-subgroups and Sylow 3-subgroups • 210:PSL(5,2) • 23+12.(S5 × PSL(3,2)) - containing Sylow 2-subgroups • U3(11):2 • M22:2 • 111+2:(5 × GL(2,3)) - normalizer of Sylow 11-subgroup • PSL(2,32):5 • • • •

PGL(2,23) U3(3) - containing Sylow 3-subgroups 29:28 = F812 43:14 = F602

240

Janko group J<sub>4 • 37:12 = F444 A Sylow 3-subgroup is a Heisenberg group: order 27, non-abelian, all non-trivial elements of order 3

References • D.J. Benson The simple group J4, PhD Thesis, Cambridge 1981, http://www.maths.abdn.ac.uk/~bensondj/ papers/b/benson/the-simple-group-J4.pdf • Ivanov, A. A. The fourth Janko group. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2004. xvi+233 pp. ISBN 0-19-852759-4 MR2124803 • Z. Janko, A new finite simple group of order 86,775,570,046,077,562,880 which possesses M24 and the full covering group of M22 as subgroups, J. Algebra 42 (1976) 564-596.doi:10.1016/0021-8693(76)90115-0 (The title of this paper is incorrect, as the full covering group of M22 was later discovered to be larger: center of order 12, not 6.) • Kleidman, Peter B.; Wilson, Robert A. (1988), "The maximal subgroups of J4" [1], Proceedings of the London Mathematical Society. Third Series 56 (3): 484–510, doi:10.1112/plms/s3-56.3.484, MR931511, ISSN 0024-6115 • S. P. Norton The construction of J4 in The Santa Cruz conference on finite groups (Ed. Cooperstein, Mason) Amer. Math. Soc 1980. • Atlas of Finite Group Representations: J4 [2]

References [1] http:/ / dx. doi. org/ 10. 1112/ plms/ s3-56. 3. 484 [2] http:/ / web. mat. bham. ac. uk/ atlas/ v2. 0/ spor/ J4/

Fischer group In mathematics, the Fischer groups are the three sporadic simple groups Fi22, Fi23,Fi24' introduced by Bernd Fischer (1971).

3-transposition groups The Fischer groups are named after Bernd Fischer who discovered them while investigating 3-transposition groups. These are groups G with the following properties: • G is generated by a conjugacy class of elements of order 2, called 'Fischer transpositions' or 3-transpositions. • The product of any two distinct transpositions has order 2 or 3. The typical example of a 3-transposition group is a symmetric group, where the Fischer transpositions are genuinely transpositions. The symmetric group Sn can be generated by n-1 transpositions: (12) ,(23), ..., (n-1,n). Fischer was able to classify 3-transposition groups that satisfy certain extra technical conditions. The groups he found fell mostly into several infinite classes (besides symmetric groups: certain classes of symplectic, unitary, and orthogonal groups), but he also found 3 very large new groups. These groups are usually referred to as Fi22, Fi23 and Fi24. The first two of these are simple groups, and the third contains the simple group Fi24' of index 2. A starting point for the Fischer groups is the unitary group PSU6(2), which could be thought of as a group Fi21 in the series of Fischer groups, of order 9,196,830,720 = 215.36.5.7.11. Actually it is the double cover 2.PSU6(2) that becomes a subgroup of the new group. This is the stabilizer of one vertex in a graph of 3510 (=2.33.5.13). These vertices become identified as conjugate 3-transpositions in the symmetry group Fi22 of the graph. The Fischer groups are named by analogy with the large Mathieu groups. In Fi22 a maximal set of 3-transpositions all commuting with one another has size 22 and is called a basic set. There are 1024 3-transpositions, called

241

Fischer group anabasic that do not commute with any in the particular basic set. Any one of other 2364, called hexadic, commutes with 6 basic ones. The sets of 6 form an S(3,6,22) Steiner system, whose symmetry group is M22. A basic set generates an abelian group of order 210, which extends in Fi22 to a subgroup 210:M22. The next Fischer group comes by regarding 2.Fi22 as a one-point stabilizer for a graph of 31671 (=34.17.23) vertices, and treating these vertices as the 3-transpositions in a group Fi23. The 3-transpositions come in basic sets of 23, 7 of which commute with a given outside 3-transposition. Next one takes Fi23 and treats it as a one-point stabilizer for a graph of 306936 (=23.33.72.29) vertices to make a group Fi24. The 3-transpositions come in basic sets of 24, 8 of which commute with a given outside 3-transposition. The group Fi24 is not simple, but its derived subgroup has index 2 and is a sporadic simple group.

Orders The order of a group is the number of elements in the group. Fi22 has order 217.39.52.7.11.13 = 64561751654400. Fi23 has order 218.313.52.7.11.13.17.23 = 4089470473293004800. Fi24' has order 221.316.52.73.11.13.17.23.29 = 1255205709190661721292800. It is the 3rd largest of the sporadic groups (after the Monster group and Baby Monster group).

Notation There is no uniformly accepted notation for these groups. Some authors use F in place of Fi (F22, for example). Fischer's notation for the them was M(22), M(23) and M(24)', which emphasised their close relationship with the three largest Mathieu groups, M22, M23 and M24. One particular source of confusion is that Fi24 is sometimes used to refer to the simple group Fi24', and is sometimes used to refer to the full 3-transposition group (which is twice the size).

References • Aschbacher, Michael (1997), 3-transposition groups (http://ebooks.cambridge.org/ebook. jsf?bid=CBO9780511759413), Cambridge Tracts in Mathematics, 124, Cambridge University Press, MR1423599, ISBN 978-0-521-57196-8 contains a complete proof of Fischer's theorem. • Fischer, Bernd (1971), "Finite groups generated by 3-transpositions. I", Inventiones Mathematicae 13: 232–246, doi:10.1007/BF01404633, MR0294487, ISSN 0020-9910 This is the first part of Fischer's preprint on the construction of his groups. The remainder of the paper is unpublished (as of 2010). • Wilson, Robert A. (2009) (in English), The finite simple groups., Graduate Texts in Mathematics 251, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-84800-988-2, Zbl: 05622792, ISBN 978-1-84800-987-5 • Wilson, R. A. "ATLAS of Finite Group Representation." http://for.mat.bham.ac.uk/atlas/html/contents.html#spo

242

Baby Monster group

Baby Monster group In the mathematical field of group theory, the Baby Monster group B (or just Baby Monster) is a group of order 241 · 313 · 56 · 72 · 11 · 13 · 17 · 19 · 23 · 31 · 47 = 4154781481226426191177580544000000 ≈ 4 · 1033. It is a simple group, meaning it does not have any normal subgroups except for the subgroup consisting only of the identity element, and B itself. The Baby Monster group is one of the sporadic groups, and has the second highest order of these, with the highest order being that of the Monster group. The double cover of the Baby Monster is the centralizer of an element of order 2 in the Monster group. The smallest faithful matrix representation of the Baby Monster is of size 4370 over the finite field of order 2. The existence of this group was suggested by Bernd Fischer in unpublished work in the early 1970s during his investigation of {3,4}-transposition groups: groups generated by a class of transpositions such that the product of any two elements has order at most 4, He investigated its properties and computed its character table; the actual construction of the Baby Monster was later realized by Jeffrey Leon and Charles Sims.[1] [2] The name "Baby Monster" was suggested by John Horton Conway[3] Höhn (1996) constructed a vertex operator algebra acted on by the baby monster, Wilson (1999) found the maximal subgroups of the baby monster. In characteristic 0 the 4372-dimensional representation of the baby monster does not have a nontrivial invariant algebra structure analogous to the Griess algebra, but Ryba (2007) showed that it does have such an invariant algebra structure if it is reduced modulo 2.

References [1] (Gorenstein 1993) [2] Leon, Jeffrey S.; Sims, Charles C. (1977). "The existence and uniqueness of a simple group generated by {3,4}-transpositions" (http:/ / projecteuclid. org/ euclid. bams/ 1183539473). Bull. Amer. Math. Soc. 83 (5): 1039–1040. . [3] Ronan, Mark (2006). Symmetry and the Monster. Oxford University Press. pp. 178–179. ISBN 0-19-280722-6.

• Gorenstein, D. (1993), "A brief history of the sporadic simple groups" (http://books.google.de/ books?id=W1TyAdpZsh8C&pg=PA141&dq=baby+monster+gruppe&hl=de& ei=l0fJTLr7BsXQ4ga62rC1Ag&sa=X&oi=book_result&ct=result&resnum=9& ved=0CE8Q6AEwCDgU#v=onepage&q&f=false), in Corwin, L.; Gelfand, I. M.; Lepowsky, James, The Gelʹfand Mathematical Seminars, 1990–1992, Boston, MA: Birkhäuser Boston, pp. 137–143, MR1247286, ISBN 978-0-8176-3689-0 • Höhn, Gerald (1996), Selbstduale Vertexoperatorsuperalgebren und das Babymonster (http://arxiv.org/abs/ 0706.0236), Bonner Mathematische Schriften [Bonn Mathematical Publications], 286, Bonn: Universität Bonn Mathematisches Institut, MR1614941 • Ryba, Alexander J. E. (2007), "A natural invariant algebra for the Baby Monster group" (http://dx.doi.org/10. 1515/JGT.2007.006), Journal of Group Theory 10 (1): 55–69, doi:10.1515/JGT.2007.006, MR2288459, ISSN 1433-5883 • Wilson, Robert A. (1999), "The maximal subgroups of the Baby Monster. I" (http://dx.doi.org/10.1006/jabr. 1998.7601), Journal of Algebra 211 (1): 1–14, doi:10.1006/jabr.1998.7601, MR1656568, ISSN 0021-8693

243

Baby Monster group

External links • MathWorld: Baby monster group (http://mathworld.wolfram.com/BabyMonsterGroup.html) • Atlas of Finite Group Representations: Baby Monster group (http://brauer.maths.qmul.ac.uk/Atlas/v3/spor/ B/)

Monster group In the mathematical field of group theory, the Monster group M or F1 (also known as the Fischer-Griess Monster, or the Friendly Giant) is a group of finite order 246 · 320 · 59 · 76 · 112 · 133 · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71 = 808017424794512875886459904961710757005754368000000000 ≈ 8 · 1053. It is a simple group, meaning it does not have any normal subgroups except for the subgroup consisting only of the identity element, and M itself. The finite simple groups have been completely classified (the classification of finite simple groups). The list of finite simple groups consists of 18 countably infinite families, plus 26 sporadic groups that do not follow such a systematic pattern. The Monster group is the largest of these sporadic groups and contains all but six of the other sporadic groups as subquotients. Robert Griess has called these six exceptions pariahs, and refers to the others as the happy family.

Existence and uniqueness The Monster was predicted by Bernd Fischer (unpublished) and Robert Griess (1976) in about 1973 as a simple group containing a double cover of Fischer's baby monster group as a centralizer of an involution. Within a few months the order of M was found by Griess using the Thompson order formula, and Fischer, Conway, Norton and Thompson discovered other groups as subquotients, including many of the known sporadic groups, and two new ones: the Thompson group and the Harada-Norton group. Griess (1982) constructed M as the automorphism group of the Griess algebra, a 196884-dimensional commutative nonassociative algebra. John Conway and Jacques Tits subsequently simplified this construction. Griess's construction showed that the Monster existed. John G. Thompson showed that its uniqueness (as a simple group of the given order) would follow from the existence of a 196883-dimensional faithful representation. A proof of the existence of such a representation was announced in 1982 by Simon P. Norton, though he has never published the details. The first published proof of the uniqueness of the Monster was completed by Griess, Meierfrankenfeld & Segev (1989). The character table of the Monster, a 194-by-194 array, was calculated in 1979 by Fischer and Livingstone using computer programs written by Thorne. The calculation was based on the assumption that the minimal degree of a faithful complex representation is 196883, which is the product of the 3 largest prime divisors of the order of M.

244

Monster group

245

Moonshine The Monster group is one of two principal constituents in the Monstrous moonshine conjecture by Conway and Norton, which relates discrete and non-discrete mathematics and was finally proved by Richard Borcherds in 1992. In this setting, the Monster group is visible as the automorphism group of the Monster module, a vertex operator algebra, an infinite dimensional algebra containing the Griess algebra, and acts on the Monster Lie algebra, a generalized Kac-Moody algebra.

McKay's E8 observation There are also connections between the monster and the extended Dynkin diagrams

specifically between the

nodes of the diagram and certain conjugacy classes in the monster, known as McKay's E8 observation.[1] [2] This is then extended to a relation between the extended diagrams and the groups 3.Fi24', 2.B, and M, where these are (3/2/1-fold central extensions) of the Fischer group, baby monster group, and monster. These are the sporadic groups associated with centralizers of elements of type 1A, 2A, and 3A in the monster, and the order of the extension corresponds to the symmetries of the diagram. See ADE classification: trinities for further connections (of McKay correspondence type), including (for the monster) with the rather small simple group PSL(2,11) and with the 120 tritangent planes of a canonic sextic curve of genus 4.

A computer construction Robert A. Wilson has found explicitly (with the aid of a computer) two 196882 by 196882 matrices (with elements in the field of order 2) which together generate the Monster group; note that this is dimension 1 lower than the 196883-dimensional representation in characteristic 0. However, performing calculations with these matrices is prohibitively expensive in terms of time and storage space. Wilson with collaborators has found a method of performing calculations with the Monster that is considerably faster. Let V be a 196882 dimensional vector space over the field with 2 elements. A large subgroup H (preferably a maximal subgroup) of the Monster is selected in which it is easy to perform calculations. The subgroup H chosen is 31+12.2.Suz.2, where Suz is the Suzuki group. Elements of the Monster are stored as words in the elements of H and an extra generator T. It is reasonably quick to calculate the action of one of these words on a vector in V. Using this action, it is possible to perform calculations (such as the order of an element of the Monster). Wilson has exhibited vectors u and v whose joint stabilizer is the trivial group. Thus (for example) one can calculate the order of an element g of the Monster by finding the smallest i > 0 such that giu = u and giv = v. This and similar constructions (in different characteristics) have been used to prove some interesting properties of the Monster (for example, to find some of its non-local maximal subgroups).

Subgroup structure

Monster group

The Monster has at least 43 conjugacy classes of maximal subgroups. Non-abelian simple groups of some 60 isomorphism types are found as subgroups or as quotients of subgroups. The largest alternating group represented is A12. The Monster contains many but not all of the 26 sporadic groups as subgroups. This diagram, based on one in the book Symmetry and the Monster by Mark Ronan, shows how they fit together. The lines signify inclusion, as a subquotient, of the lower group by the upper one. The circled symbols denote groups not involved in larger sporadic groups. For the sake of clarity redundant inclusions are not shown.

246

Sporadic Finite Groups Showing (Sporadic) Subgroups

Occurrence The monster can be realized as a Galois group over the rational numbers (Thompson 1984, p. 443), and as a Hurwitz group (Wilson 2004).

Notes [1] Arithmetic groups and the affine E8 Dynkin diagram (http:/ / arxiv4. library. cornell. edu/ abs/ 0810. 1465), by John F. Duncan, in Groups and symmetries: from Neolithic Scots to John McKay [2] le Bruyn, Lieven (22 April 2009), the monster graph and McKay’s observation (http:/ / www. neverendingbooks. org/ index. php/ the-monster-graph-and-mckays-observation. html),

References • J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. London Math. Soc. 11 (1979), no. 3, 308—339. • Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; and Wilson, R. A.: Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups. Oxford, England 1985. • Griess, Robert L. (1976), "The structure of the monster simple group", in Scott, W. Richard; Gross, Fletcher, Proceedings of the Conference on Finite Groups (Univ. Utah, Park City, Utah, 1975), Boston, MA: Academic Press, pp. 113–118, MR0399248, ISBN 978-0-12-633650-4 • Griess, Robert L. (1982), "The friendly giant", Inventiones Mathematicae 69 (1): 1–102, doi:10.1007/BF01389186, MR671653, ISSN 0020-9910 • Griess, Robert L; Meierfrankenfeld, Ulrich; Segev, Yoav (1989), "A uniqueness proof for the Monster" (http:// jstor.org/stable/1971455), Annals of Mathematics. Second Series 130 (3): 567–602, doi:10.2307/1971455, MR1025167, ISSN 0003-486X • Harada, Koichiro (2001), "Mathematics of the Monster", Sugaku Expositions 14 (1): 55–71, MR1690763, ISSN 0898-9583 • P. E. Holmes and R. A. Wilson, A computer construction of the Monster using 2-local subgroups, J. London Math. Soc. 67 (2003), 346—364. • Ivanov, A. A., The Monster Group and Majorana Involutions, Cambridge tracts in mathematics, 176, Cambridge University Press, ISBN 978-0521889940 • S. A. Linton, R. A. Parker, P. G. Walsh and R. A. Wilson, Computer construction of the Monster, J. Group Theory 1 (1998), 307-337. • S. P. Norton, The uniqueness of the Fischer-Griess Monster, Finite groups---coming of age (Montreal, Que., 1982), 271—285, Contemp. Math., 45, Amer. Math. Soc., Providence, RI, 1985. • M. Ronan, Symmetry and the Monster, Oxford University Press, 2006, ISBN 0192807226 (concise introduction for the lay reader).

Monster group • M. du Sautoy, Finding Moonshine, Fourth Estate, 2008, ISBN 978-0-00-721461-7 (another introduction for the lay reader; published in the US by HarperCollins as Symmetry, ISBN 978-0060789404). • Thompson, John G. (1984), "Some finite groups which appear as Gal L/K, where K ⊆ Q(μn)", Journal of Algebra 89 (2): 437–499, doi:10.1016/0021-8693(84)90228-X, MR751155. • Wilson, Robert A. (2001), "The Monster is a Hurwitz group" (http://web.mat.bham.ac.uk/R.A.Wilson/pubs/ MHurwitz.ps), Journal of Group Theory 4 (4): 367–374, doi:10.1515/jgth.2001.027, MR1859175

External links • MathWorld: Monster Group (http://mathworld.wolfram.com/MonsterGroup.html) • Atlas of Finite Group Representations: Monster group (http://brauer.maths.qmul.ac.uk/Atlas/v3/spor/M/) • Abstruse Goose: Fischer-Griess Monster (http://abstrusegoose.com/96)

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 Source: http://en.wikipedia.org/w/index.php?oldid=401806125  Contributors: AndrewKepert, Anonymous Dissident, Auximines, AxelBoldt, Beland, BenFrantzDale, Bornintheguz, Charles Matthews, Conversion script, Cullinane, Debivort, Dominus, Dysprosia, Eubulides, Fadereu, Fropuff, Giftlite, Hannes Eder, IstvanWolf, Jkominek, Josh Grosse, KSmrq, Kilva, LarryLACa, Lillebi, MFH, Martin von Gagern, Maverick starstrider, Mets501, Minimac's Clone, Nonenmac, Oleg Alexandrov, Patrick, Pcgomes, Qutezuce, Raven4x4x, RedWolf, Reverendgraham, Rich Farmbrough, RobinK, Romaioi, Siddhant, Silly rabbit, Sir Vicious, Snags, Stevenj, Stevertigo, Sverdrup, Tamfang, Tarquin, TheLimbicOne, TimBentley, Tobias Bergemann, Tokek, Wikipedist, 43 anonymous edits Symmetric group  Source: http://en.wikipedia.org/w/index.php?oldid=408788612  Contributors: A8UDI, Akriasas, Am.hussein, Andre Engels, AnnaFrance, Arcfrk, AxelBoldt, BjornPoonen, CBM, CRGreathouse, Charles Matthews, Conversion script, Damian Yerrick, Doctorhook, Dogah, Dysprosia, Eighthdimension, Erud, Ezrakilty, Fredrik, Giftlite, Goochelaar, GraemeMcRae, Graham87, Grubber, Hayabusa future, Helder.wiki, Huppybanny, Icekiss, JackSchmidt, Jim.belk, Jirka62, Kingpin13, LarryLACa, Linas, Looxix, MFH, MSchmahl, Mandarax, Marc van Leeuwen, Mhym, Michael Hardy, Michael Slone, Mpatel, Mzamora2, NatusRoma, Nbarth, Ojigiri, Paradoxsociety, Patrick, Paul August, Paul Matthews, Pcap, Pcgomes, Phys, PierreAbbat, Pred, Rayk1212, Rubybrian, Salix alba, Sandrobt, Simetrical, SirJective, Stifle, Tamfang, Tobias Bergemann, Tosha, Wohingenau, Wshun, Zero0000, Zundark, 62 anonymous edits Combinatorial group theory  Source: http://en.wikipedia.org/w/index.php?oldid=366117538  Contributors: CBM, Cambyses, Charles Matthews, Gvozdet, JackSchmidt, Nbarth Algebraic group  Source: http://en.wikipedia.org/w/index.php?oldid=393322595  Contributors: AxelBoldt, Bprsolt Qaoddz, Charles Matthews, Cronholm144, David Eppstein, DeaconJohnFairfax, Dysprosia, Fropuff, Giftlite, Hesam7, JackSchmidt, Jakob.scholbach, Jim.belk, Joerg Winkelmann, Krasnoludek, Linas, LokiClock, Michael Hardy, Michael Kinyon, Nbarth, Paul August, Ppntori, R.e.b., TakuyaMurata, Turgidson, Vivacissamamente, Waltpohl, 8 anonymous edits Solvable group  Source: http://en.wikipedia.org/w/index.php?oldid=406524716  Contributors: 99 Willys on Wheels on the wall, 99 Willys on Wheels..., AxelBoldt, Badanedwa, Bird of paradox, Charles Matthews, Chas zzz brown, Cícero, DYLAN LENNON, Dogah, Dogaroon, Dr Zimbu, Dysprosia, ElNuevoEinstein, Fibonacci, Fropuff, Gandalfxviv, Gfis, Giftlite, Golbez, JackSchmidt, Jakob.scholbach, Jeni, Jim.belk, Jweimar, Kilva, Lausailuk, Lifthrasir1, Lupin, Malcolm Farmer, MathMartin, Michael Hardy, Mlpearc, Nbarth, Paddles, Patrick, Phys, R.e.b., RadioActive, Schildt.a, Seb35, Stewartadcock, Tobias Bergemann, Tosha, Turgidson, Vaughan Pratt, Vipul, Weregerbil, Zundark, 38 anonymous edits Solvable subgroup  Source: http://en.wikipedia.org/w/index.php?oldid=17229888  Contributors: 99 Willys on Wheels on the wall, 99 Willys on Wheels..., AxelBoldt, Badanedwa, Bird of paradox, Charles Matthews, Chas zzz brown, Cícero, DYLAN LENNON, Dogah, Dogaroon, Dr Zimbu, Dysprosia, ElNuevoEinstein, Fibonacci, Fropuff, Gandalfxviv, Gfis, Giftlite, Golbez, JackSchmidt, Jakob.scholbach, Jeni, Jim.belk, Jweimar, Kilva, Lausailuk, Lifthrasir1, Lupin, Malcolm Farmer, MathMartin, Michael Hardy, Mlpearc, Nbarth, Paddles, Patrick, Phys, R.e.b., RadioActive, Schildt.a, Seb35, Stewartadcock, Tobias Bergemann, Tosha, Turgidson, Vaughan Pratt, Vipul, Weregerbil, Zundark, 38 anonymous edits Tits building  Source: http://en.wikipedia.org/w/index.php?oldid=158816238  Contributors: Arcfrk, Charles Matthews, Chenxlee, D6, David Eppstein, DavidCBryant, Delirium, Giftlite, J.delanoy, Jon Awbrey, Joseph Myers, Julyo, KRS, Lantonov, MSGJ, Mathsci, Mhym, Michael Hardy, Mr Adequate, N5iln, Oleg Alexandrov, Omg wtf lol stfu noob, Omnipaedista, R.e.b., RUL3R, Rjwilmsi, Rror, Sdfgtsryedry124214, Stwitzel, Tango, Tide rolls, Trovatore, 13 anonymous edits Finite group  Source: http://en.wikipedia.org/w/index.php?oldid=408239689  Contributors: ABCD, Alberto da Calvairate, Andi5, AxelBoldt, Baccyak4H, Charles Matthews, Ciphers, Cullinane, D3, DHN, Dreadstar, Geometry guy, Giftlite, HenryLi, JackSchmidt, Kilva, LGB, Loren Rosen, Messagetolove, Mhym, Michael Hardy, Oleg Alexandrov, Patrick, Phys, R.e.b., Radagast3, Rgdboer, Schneelocke, Schutz, Silverfish, SparsityProblem, TakuyaMurata, Thehotelambush, Vipul, Zundark, 42 anonymous edits p-adic number  Source: http://en.wikipedia.org/w/index.php?oldid=402961522  Contributors: 130.182.125.xxx, A5, Adam majewski, Arthur Rubin, AxelBoldt, Ben Standeven, Bender235, Bluap, Brentt, Bryan Derksen, CRGreathouse, Charles Matthews, Chas zzz brown, Chinju, Chowbok, Chris the speller, Ciphergoth, Classicalecon, Codygunton, Conversion script, Coopercc, CryptoDerk, DFRussia, Damian Yerrick, David Eppstein, Dcoetzee, DeaconJohnFairfax, Dharma6662000, Dnas, Dominus, Dratman, Drusus 0, Dysprosia, Długosz, E.V.Krishnamurthy, Eequor, ElNuevoEinstein, Emurphy42, Eric Drexler, Eric Kvaalen, Fropuff, Gandalf61, Gauge, Gene Ward Smith, Giftlite, Graham87, H00kwurm, Hairchrm, Hans Adler, Haziel, Heptadecagon, Ideyal, Ilanpi, Iseeaboar, Isnow, JackSchmidt, Jafet, Jallotta, Jbolden1517, JeffBurdges, KSmrq, Keith Edkins, Kier07, Kusma, Lambiam, Lethe, Linas, Looxix, MFH, MarSch, Marozols, MathMartin, Mav, Maxal, Melchoir, Michael Hardy, Miguel, Mikolt, Minesweeper, Mon4, Nbarth, Oleg Alexandrov, Oli Filth, Patrick, Paul August, PaulTanenbaum, PierreAbbat, Pjacobi, Populus, Qpt, ReiVaX, Revolver, Rill2503456, Rotem Dan, RxS, Singingwolfboy, SirJective, Sligocki, Stephen Bain, Tachyon², Taejo, TakuyaMurata, Thatcher, The Anome, TheBlueWizard, Toby Bartels, Tosha, Trovatore, Wadems, Waltpohl, Zundark, 118 anonymous edits Tits alternative  Source: http://en.wikipedia.org/w/index.php?oldid=402301072  Contributors: Charles Matthews, DavidCBryant, JackSchmidt, Jim.belk, Nsk92, Sdfgsgedy454, Snigbrook, 3 anonymous edits

248

Article Sources and Contributors Finitely generated group  Source: http://en.wikipedia.org/w/index.php?oldid=54295413  Contributors: ArnoldReinhold, Artem M. Pelenitsyn, AxelBoldt, CRGreathouse, Charles Matthews, Chas zzz brown, Chinju, Dbenbenn, Dcoetzee, Dr.enh, Dysprosia, Emperorbma, Eyal0, Fibonacci, Giftlite, Herbee, JackSchmidt, Lenthe, Mhss, Michael Hardy, Michael Slone, Optimisteo, RobHar, Romanm, Tomo, Vp loreta, Zundark, 14 anonymous edits Linear group  Source: http://en.wikipedia.org/w/index.php?oldid=246251120  Contributors: 3children, Anterior1, Arcfrk, AxelBoldt, Charles Matthews, Ikh, JackSchmidt, KSmrq, Keyi, Malcolmxl5, MatrixHugh, Michael Hardy, NarrabundahMan, Ndbrian1, R.e.b., RHB, Salix alba, TooMuchMath, Vanished User 0001, Zaslav, 5 anonymous edits Finite index  Source: http://en.wikipedia.org/w/index.php?oldid=280259017  Contributors: 4pq1injbok, AxelBoldt, Danramras, Doody.parizada, Druiffic, Giftlite, JackSchmidt, Jim.belk, Koavf, Mathsci, Nbarth, Quotient group, Si biskuit, Tobias Bergemann, 5 anonymous edits Free subgroup  Source: http://en.wikipedia.org/w/index.php?oldid=71154285  Contributors: ATC2, Altenmann, Archelon, AxelBoldt, C S, Charles Matthews, Chris Pressey, Dbenbenn, Dysprosia, Fadereu, Giftlite, HenrikRueping, Hyginsberg, Iorsh, JackSchmidt, Jim.belk, Kapitolini, Kidburla, LachlanA, Larsbars, Laurentius, Linas, Marozols, MathMartin, Mathsci, Mct mht, Michael Hardy, Mikeblas, Mohan ravichandran, Punainen Nörtti, R.e.b., Ralamosm, Reedy, Rjwilmsi, Robert Illes, RonnieBrown, Sam nead, Tiphareth, Tobias Bergemann, Tomo, Tosha, Trovatore, Turgidson, Vipul, Virginia-American, ZeroOne, Ziyuang, Zundark, Мыша, 31 anonymous edits Tits group  Source: http://en.wikipedia.org/w/index.php?oldid=397111631  Contributors: Alchemist Jack, Alison, Baseball Bugs, Bishi Bosche, Bkell, Boemmels, Buster79, Catgut, Charles Matthews, Chzz, David.Monniaux, Edman1959, Frehley, Gene Ward Smith, Ginsengbomb, Grafen, Huppybanny, Imo1234, JackSchmidt, Jmmuguerza, Michael Hardy, Michael Slone, MuZemike, Oleg Alexandrov, Pyrop, R.e.b., Sietse Snel, Silverfish, Smjg, SoSaysChappy, Srd2005, ThanksForTheFish, 47 anonymous edits Tits–Koecher construction  Source: http://en.wikipedia.org/w/index.php?oldid=404738351  Contributors: R.e.b. Primitive group  Source: http://en.wikipedia.org/w/index.php?oldid=23861834  Contributors: Charles Matthews, Cullinane, Dvorak729, Dysprosia, Keenan Pepper, Michael Kinyon, Paul August, R.e.b., Richard L. Peterson, Stefan Kohl, Turgidson, 3 anonymous edits Geometric group theory  Source: http://en.wikipedia.org/w/index.php?oldid=394978510  Contributors: Artem M. Pelenitsyn, C S, Cambyses, Charles Matthews, Chris the speller, Dancter, Dbenbenn, Frazzydee, Fropuff, Giftlite, JackSchmidt, Jevansen, Jheald, Jim.belk, Juan Marquez, LarRan, MathMartin, Mboverload, Michael Hardy, Nbarth, Nsk92, Oleg Alexandrov, Reiner Martin, Rjwilmsi, Silverfish, Staecker, SunCreator, Turgidson, Wireader, 12 anonymous edits Hyperbolic group  Source: http://en.wikipedia.org/w/index.php?oldid=401515762  Contributors: Arcfrk, C S, Ceyockey, Charles Matthews, Charvest, Dbenbenn, Gauge, Giftlite, Haroldsultan, Ikapovitch, JackSchmidt, Jevansen, LarRan, Mad2Physicist, Ptreth, Quotient group, TheAstonishingBadger, Thefrettinghand, Tosha, Turgidson, Yottie, Zundark, 13 anonymous edits Automatic group  Source: http://en.wikipedia.org/w/index.php?oldid=384649636  Contributors: AutomatonTheorist, C S, Charles Matthews, Charvest, David Eppstein, Dysprosia, Gauge, Jka02, JoshuaZ, Michael Hardy, NawlinWiki, Quotient group, Rjwilmsi, Samuel Blanning, Vipul, 5 anonymous edits Discrete group  Source: http://en.wikipedia.org/w/index.php?oldid=399469544  Contributors: Arcfrk, Cambyses, Charles Matthews, Dreadstar, Fropuff, Giftlite, Jakob.scholbach, Jim.belk, JoergenB, Linas, Maksim-e, Mhss, Michael Hardy, Mosher, RDBury, Topology Expert, Zundark, 9 anonymous edits Todd–Coxeter algorithm  Source: http://en.wikipedia.org/w/index.php?oldid=396543603  Contributors: Andreas Kaufmann, Arcfrk, Bkonrad, Booyabazooka, CBM, Calliopejen1, Charles Matthews, Dtrebbien, Dysprosia, JackSchmidt, Rswarbrick, Superninja, Taxiarchos228, 4 anonymous edits Frobenius group  Source: http://en.wikipedia.org/w/index.php?oldid=397433656  Contributors: Charles Matthews, Fropuff, Giftlite, I dream of horses, JackSchmidt, Jdgilbey, Jim.belk, Mathsci, Michael Hardy, R.e.b., Rl, 15 anonymous edits Zassenhaus group  Source: http://en.wikipedia.org/w/index.php?oldid=356315687  Contributors: Charles Matthews, Everyking, JackSchmidt, Jim.belk, Michael Hardy, Nbarth, R.e.b., 1 anonymous edits Regular p-group  Source: http://en.wikipedia.org/w/index.php?oldid=399150318  Contributors: JackSchmidt, Jim.belk, Michael Hardy, R.e.b., Wikiadamg, Zundark, 4 anonymous edits Isoclinism of groups  Source: http://en.wikipedia.org/w/index.php?oldid=402483364  Contributors: JackSchmidt, Michael Hardy, Richard L. Peterson, 1 anonymous edits Variety (universal algebra)  Source: http://en.wikipedia.org/w/index.php?oldid=390624927  Contributors: Backslash Forwardslash, Cambyses, Charles Matthews, Chuunen Baka, Dorchard, Giftlite, Hans Adler, JMK, JackSchmidt, Jesper Carlstrom, LilHelpa, Linas, Livajo, Michael Hardy, Nbarth, Pascal.Tesson, Pavel Jelinek, Smimram, Taeshadow, Thorwald, Tobias Bergemann, Trovatore, Uncle G, Untalker, Vaughan Pratt, Woohookitty, Zoz, 9 anonymous edits Reflection group  Source: http://en.wikipedia.org/w/index.php?oldid=363727802  Contributors: Arcfrk, Biruitorul, Charles Matthews, Chuckrocks, Cullinane, EagleFan, Frankchn, Giftlite, Jim.belk, Johnpseudo, KSmrq, Kuru, Melchoir, Mxn, Nbarth, Nopetro, Oleg Alexandrov, Patrick, Pseudomonas, R.e.b., Riana, Sławomir Biały, Vyznev Xnebara, 14 anonymous edits Fundamental group  Source: http://en.wikipedia.org/w/index.php?oldid=393898587  Contributors: Akriasas, Alksentrs, Andi5, Archelon, AxelBoldt, Blotwell, Cbigorgne, Charles Matthews, Conversion script, Cruccone, Dan Gardner, Dpv, Dr Dec, Dysprosia, ElNuevoEinstein, Fropuff, Gauge, Giftlite, Haiviet, Hans Adler, HiDrNick, Hirak 99, Ht686rg90, JackSchmidt, Jakob.scholbach, Jim.belk, Klausness, KonradVoelkel, Lethe, Linas, Mathsci, Michael Hardy, Msh210, Myasuda, Nbarth, OdedSchramm, Oerjan, Orthografer, Patrick, Phys, Point-set topologist, Poor Yorick, R.e.b., Ranicki, Raven in Orbit, Rgrizza, Ringspectrum, Rjwilmsi, Sam nead, Senouf, Silly rabbit, Staecker, TakuyaMurata, The Thing That Should Not Be, Tobias Bergemann, Tosha, Turgidson, Wlod, Zundark, 39 anonymous edits Classical group  Source: http://en.wikipedia.org/w/index.php?oldid=401841571  Contributors: Arcfrk, Charles Matthews, Gareth McCaughan, Krasnoludek, Nbarth, Pt, R.e.b., RobHar, Semorrison, 6 anonymous edits Unitary group  Source: http://en.wikipedia.org/w/index.php?oldid=404556311  Contributors: Aghitza, AxelBoldt, CXCV, Charles Matthews, Dr Zimbu, Drschawrz, Fropuff, Giftlite, HappyCamper, JATerg, JackSchmidt, JarahE, Jjalexand, Keyi, KnightRider, Linas, Looxix, MarSch, Michael Hardy, Nbarth, Niout, R.e.b., RobHar, Ruud Koot, Silly rabbit, Winston365, Yartsa, Zundark, 18 anonymous edits Character theory  Source: http://en.wikipedia.org/w/index.php?oldid=398197335  Contributors: Alan smithee, Alecobbe, Arcfrk, Ashsong, BlackFingolfin, Bobo192, Charles Matthews, Crink, Cweaton, Eric Kvaalen, FelixP, Francs2000, Frau Holle, Fropuff, Geffrey, Giftlite, Grubber, Hesam7, Hillman, Icairns, Jim.belk, Jtwdog, Kilva, Lethe, Linas, MathMartin, Messagetolove, Michael Kinyon, MultimediaGuru, Nbarth, Numenorean7, Oleg Alexandrov, PROUDKEEP, Paul Matthews, Phys, Point-set topologist, Qutezuce, R'n'B, R.e.b., Ringspectrum, Rjwilmsi, RobHar, Snags, Sullivan.t.j, Swift chlr, Tesseran, Xiaodai, 16 anonymous edits Sylow theorem  Source: http://en.wikipedia.org/w/index.php?oldid=98222564  Contributors: 01001, Aholtman, Amitushtush, Ams80, Ank0ku, AxelBoldt, BenF, BeteNoir, CZeke, Charles Matthews, Chas zzz brown, Chochopk, Conversion script, Crisófilax, Cwkmail, David Eppstein, Derek Ross, Dominus, Druiffic, EmilJ, Eramesan, Functor salad, GTBacchus, Gauge, Geometry guy, Giftlite, Goochelaar, Graham87, Grubber, Haham hanuka, Hank hu, Hesam7, JackSchmidt, Japanese Searobin, Joelsims80, Jonathanzung, Kilva, Lzur, MathMartin, Mav, Michael Hardy, Nbarth, Ossido, PierreAbbat, Pladdin, Pmanderson, Point-set topologist, Pyrop, R.e.b., Reedy, Schutz, Siroxo, Sl, Spoon!, Stove Wolf, Superninja, TakuyaMurata, Tarquin, Tobias Bergemann, Twilsonb, WLior, Welsh, Zundark, Zvika, 75 anonymous edits Lie algebra  Source: http://en.wikipedia.org/w/index.php?oldid=392789870  Contributors: Adam cohenus, AlainD, Arcfrk, Arthena, Asimy, AxelBoldt, BenFrantzDale, Bogey97, CSTAR, Chameleon, Charles Matthews, Conversion script, CryptoDerk, Curps, Dachande, David Gerard, DefLog, Drbreznjev, Drorata, Dysprosia, Englebert, Foobaz, Freiddie, Fropuff, Gauge, Geometry guy, Giftlite, Grendelkhan, Grokmoo, Grubber, Gvozdet, Hairy Dude, Harold f, Hesam7, Iorsh, Isnow, JackSchmidt, Jason Quinn, Jason Recliner, Esq., Jeremy Henty, Jkock, Joel Koerwer, [email protected], Juniuswikiae, Kaoru Itou, Kragen, Kwamikagami, Lenthe, Lethe, Linas, Loren Rosen, MarSch, Masnevets, Michael Hardy, Michael Larsen, Michael Slone, Miguel, Msh210, NatusRoma, Nbarth, Ndbrian1, Niout, Noegenesis, Oleg Alexandrov, Paolo.dL, Phys, Pizza1512, Pj.de.bruin, Prtmrz, Pt, Pyrop, Python eggs, R'n'B, Reinyday, RexNL, Rossami, Sbyrnes321, Shirulashem, Silly rabbit, Spangineer, StevenJohnston, Suisui, Supermanifold, TakuyaMurata, Thomas Bliem, Tobias Bergemann, Tosha, Twri, Vanish2, Veromies, Wavelength, Weialawaga, Wood Thrush, Wshun, Zundark, 84 anonymous edits Class group  Source: http://en.wikipedia.org/w/index.php?oldid=16844210  Contributors: Alodyne, AxelBoldt, CRGreathouse, Charles Matthews, Danpovey, DeaconJohnFairfax, Dmharvey, Dyss, Gauge, Gene Ward Smith, Georg Muntingh, Giftlite, Grubber, Hesam7, Ilion2, Michael Hardy, Mon4, Nbarth, Pmanderson, PoolGuy, RobHar, Roentgenium111, Smjwilson, TakuyaMurata, Timwi, Tobias Bergemann, Virginia-American, Vivacissamamente, Waltpohl, Wshun, Zundark, 18 anonymous edits Abelian group  Source: http://en.wikipedia.org/w/index.php?oldid=403828404  Contributors: 128.111.201.xxx, Aeons, Amire80, Andres, Andyparkerson, Arcfrk, AxelBoldt, Brighterorange, Brona, Bryan Derksen, CRGreathouse, Charles Matthews, Chas zzz brown, Chowbok, Ciphers, Coleegu, Conversion script, DHN, DL144, Dcoetzee, Doradus, Dr Caligari, Drbreznjev,

249

Article Sources and Contributors Drgruppenpest, Drilnoth, Dysprosia, Fibonacci, Fropuff, GB fan, Gandalf61, Gauge, Geschichte, Giftlite, Gregbard, Grubber, Helder.wiki, Isnow, JackSchmidt, Jdforrester, Jitse Niesen, Jlaire, Joe Campbell, Johnuniq, Jonathans, Jorend, Kaoru Itou, Karada, Keenan Pepper, Konradek, Lagelspeil, Leonard G., Lethe, Lovro, Madmath789, Magic in the night, Mathisreallycool, Mets501, Michael Hardy, Michael Slone, Mikael V, Namwob0, Negi(afk), Newone, Oleg Alexandrov, Oli Filth, Pakaran, Patrick, Philosophygeek, Pmanderson, Poor Yorick, Quotient group, R.e.b., Recognizance, Revolver, Rickterp, Romanm, Salix alba, Saxbryn, Schneelocke, SetaLyas, Shenme, Silly rabbit, SirJective, Ste4k, Stevertigo, Stifle, TakuyaMurata, Tango, Theresa knott, Tobias Bergemann, Topology Expert, Trhaynes, Vanish2, Vaughan Pratt, Vipul, Waltpohl, Warut, Zabadooken, Zundark, 90 anonymous edits Lie group  Source: http://en.wikipedia.org/w/index.php?oldid=408761117  Contributors: 212.29.241.xxx, Abdull, Akriasas, Alex Varghese, AnmaFinotera, Anterior1, Arcfrk, Archelon, Arkapravo, AxelBoldt, BMF81, Badger014, Barak, Bears16, Beastinwith, Beland, BenFrantzDale, Benjamin.friedrich, Bobblewik, Bongwarrior, Borat fan, Brian Huffman, Buster79, CBM, CRGreathouse, Cacadril, Charles Matthews, Cherlin, ChrisJ, Cmelby, Conversion script, Dablaze, Darkfight, Davewild, David Eppstein, David Shay, DefLog, Dorftrottel, Dr.enh, Drorata, Dysprosia, Dzordzm, Ekeb, Eubulides, FlashSheridan, Fropuff, GTBacchus, Genuine0legend, Geometry guy, Giftlite, Graham87, HappyCamper, Headbomb, Hesam7, Hillman, Homeworlds, Ht686rg90, Inquisitus, Isnow, Itai, JDspeeder1, JackSchmidt, James.r.a.gray, JamesMLane, Jason Quinn, Jesper Carlstrom, Jim.belk, Jitse Niesen, Joriki, Josh Cherry, Josh Grosse, JustAGal, KSmrq, Kaoru Itou, KbReZiE 12, KeithB, Kier07, Krasnoludek, Kwamikagami, Len Raymond, Leontios, Lethe, Linas, Lockeownzj00, Logical2u, Looxix, Lseixas, MarSch, Marc van Leeuwen, Masnevets, Mathchem271828, Mhss, Michael Hardy, Michael Kinyon, Michael Slone, Miguel, MotherFunctor, Msh210, MuDavid, Myasuda, NatusRoma, Ndbrian1, Ninte, Niout, Oleg Alexandrov, Orthografer, Oscarbaltazar, Ozob, PAR, Paul August, Phys, Pidara, Pmanderson, Porcher, Pred, R.e.b., Rgdboer, RichardVeryard, RobHar, Rocket71048576, RodVance, S2000magician, Saaska, Salgueiro, Salix alba, Shanes, Sidiropo, Silly rabbit, Siva1979, Smaines, Smylei, Stevertigo, Sullivan.t.j, Suslindisambiguator, Sławomir Biały, Tanath, Tide rolls, Tobias Bergemann, Tom Lougheed, Tompw, TomyDuby, Topology Expert, Tosha, Trevorgoodchild, Ulner, Unifey, VKokielov, Weialawaga, Wgmccallum, WhatamIdoing, XJamRastafire, Xantharius, Xavic69, Yggdrasil014, Zoicon5, Zundark, 111 anonymous edits Galois group  Source: http://en.wikipedia.org/w/index.php?oldid=389520137  Contributors: Alro, AugPi, AxelBoldt, Charles Matthews, Chowbok, Conversion script, Cwkmail, Dan Gardner, Daniel Mahu, Dmharvey, Dyaa, Dysprosia, EmilJ, Fredrik, Giftlite, Grubber, Helder.wiki, Hesam7, JackSchmidt, Jakob.scholbach, Keyi, Lagelspeil, Loren Rosen, MattTait, Michael Hardy, Moxmalin, Point-set topologist, RobHar, TakuyaMurata, TomyDuby, Unyoyega, Vivacissamamente, Zundark, 16 anonymous edits General linear group  Source: http://en.wikipedia.org/w/index.php?oldid=399571821  Contributors: A5, Albmont, AxelBoldt, Charles Matthews, Chas zzz brown, Cullinane, Dmharvey, Drschawrz, Dysprosia, EmilJ, Franp9am, Fropuff, Gaius Cornelius, Gauge, Giftlite, Goudzovski, Greenfernglade, Gwaihir, HappyCamper, Harryboyles, Ht686rg90, Huppybanny, JackSchmidt, Jeepday, Jim.belk, Jitse Niesen, KSmrq, KnightRider, Linas, Llanowan, MSGJ, Marconet, Mhss, Michael Hardy, Michael Slone, Msh210, Nbarth, Niout, Oleg Alexandrov, Patrick, Paul August, Pleasantville, R.e.b., Salix alba, Silly rabbit, Spvo, Sullivan.t.j, Topology Expert, Weialawaga, Zero sharp, Zhaoway, Zundark, 41 anonymous edits Representation theory  Source: http://en.wikipedia.org/w/index.php?oldid=407178563  Contributors: Andresswift, BenFrantzDale, CBM, Cyfal, Frau Holle, Geometry guy, Giftlite, Hugh16, KathrynLybarger, Kiefer.Wolfowitz, Mild Bill Hiccup, PaulTanenbaum, Pred, R'n'B, RobHar, The Thing That Should Not Be, Unfree, Wavelength, Zundark, 18 anonymous edits Symmetry in physics  Source: http://en.wikipedia.org/w/index.php?oldid=328220556  Contributors: 8af4bf06611c, A. di M., AndrewHowse, Archelon, BenFrantzDale, Bloodshedder, Bradv, Brews ohare, Christian75, Commander Keane, Complexica, Danski14, Divey, Email4mobile, Fratrep, Giftlite, Heron, Homunq, Hotbody, JRSpriggs, Janus Shadowsong, Joshua P. Schroeder, Manganite, Mattpickman, Mets501, Michael C Price, Michael Hardy, Mpatel, Netoholic, Oleg Alexandrov, Ottre, PV=nRT, Paradoctor, Patrick, Paul D. Anderson, PhilKnight, Physicistjedi, Point-set topologist, Quodfui, Reaverdrop, Rorro, Rror, Sbyrnes321, Stevertigo, StradivariusTV, Stylus881, Thamuzino, The Anome, Woohookitty, X42bn6, YK Times, 45 anonymous edits Space group  Source: http://en.wikipedia.org/w/index.php?oldid=395242094  Contributors: 2over0, Ambarsande, Asrghasrhiojadrhr, Baccyak4H, Bwmodular, Cbup, Charles Matthews, DeadEyeArrow, Dmb000006, Egalegal, Encephalon, Felipe Gonçalves Assis, Giftlite, Hetar, Jaccos, JackSchmidt, Jcwf, Jimduck, Joseph Myers, KSmrq, Michael Hardy, Mpatel, Nikai, Oleg Alexandrov, Oysteinp, Patrick, Polyamorph, R.e.b., Rifleman 82, Rjwilmsi, Rossami, Soc8675309, Syntax, Tagishsimon, Tantalate, Template namespace initialisation script, That Guy, From That Show!, Tomruen, Tosha, Truelight, Vespristiano, Vsmith, Warp0009, Wik, WikHead, Yhshin, Սահակ, 68 anonymous edits Molecular symmetry  Source: http://en.wikipedia.org/w/index.php?oldid=401210282  Contributors: Baccyak4H, Bdevill, Benjah-bmm27, Benjaminruggill, Bit Lordy, Bobby1011, Crystal whacker, DeadEyeArrow, Dirac66, Gamingmaster125, Hongooi, Itub, Kelix, Kero584, Kindofply, L Kensington, Nono64, Paolo.dL, Pentalis, Petronas, Ph0987, RAWAL SANJAY, Rjwilmsi, Schmloof, Smokefoot, SpaceFlight89, Thegeneralguy, V8rik, Zargulon, 25 anonymous edits Applications of group theory  Source: http://en.wikipedia.org/w/index.php?oldid=200345224  Contributors: Adan, Adgjdghjdety, Alberto da Calvairate, Ale jrb, Alksentrs, Alpha Beta Epsilon, Arcfrk, Archie Paulson, ArnoldReinhold, ArzelaAscoli, Auclairde, Avouac, AxelBoldt, Baccyak4H, Bevo, Bhuna71, BiT, Bogdangiusca, Bongwarrior, CRGreathouse, Calcio33, Cate, Cessator, Charles Matthews, Chris Pressey, Chun-hian, Cmbankester, ComplexZeta, CountingPine, Cwitty, CàlculIntegral, D stankov, D15724C710N, DYLAN LENNON, David Callan, David Eppstein, Davipo, Dcljr, Debator of mathematics, Dennis Estenson II, Doshell, Dratman, Drschawrz, Dysprosia, Eakirkman, Eamonster, EchoBravo, Edward, Edwinconnell, Eubulides, Favonian, Fibonacci, Finlay McWalter, Friviere, GBL, Gabriel Kielland, Gandalf61, Giftlite, Gombang, Googl, Graeme Bartlett, Gregbard, Gromlakh, Grubber, H00kwurm, Hairy Dude, Hamtechperson, Hans Adler, HenryLi, Hillman, Hyacinth, Indeed123, Ivan Štambuk, J.delanoy, JWSchmidt, JackSchmidt, Jaimedv, Jakob.scholbach, Jauhienij, JinJian, Jitse Niesen, Jordi Burguet Castell, Josh Parris, Justin W Smith, KF, Karl-Henner, Kristine8, Kwantus, Lambiam, Lemonaftertaste, Lfh, LiDaobing, Lightmouse, Ligulem, Lipedia, Luqui, M cuffa, MTC, MaEr, Maedin, Magmi, Manuel Trujillo Berges, Masv, MathMartin, Mayooranathan, Merlincooper, Messagetolove, Michael Hardy, Michael Slone, Mike Fikes, Mspraveen, NERIUM, Nadav1, Natebarney, Ngyikp, NobillyT, Obradovic Goran, OdedSchramm, Orhanghazi, Padicgroup, Papadopc, Paul August, Peter Stalin, PeterPearson, Petter Strandmark, Philip Trueman, Phys, Pieter Kuiper, Pilotguy, Poor Yorick, R.e.b., Ranveig, Recentchanges, Reedy, Rich Farmbrough, Richard L. Peterson, Rifleman 82, Rjwilmsi, RobHar, Romanm, RonnieBrown, Rossami, Rune.welsh, Rursus, Salix alba, Scullin, SomeRandomPerson23, Sławomir Biały, Tbsmith, The Anome, Tigershrike, TimothyRias, Tommy2010, Tompw, Tyskis, Useight, Utopianheaven, V8rik, Vegetator, VictorAnyakin, Viskonsas, WVhybrid, Willtron, WinoWeritas, Wshun, Xylthixlm, Yger, Zundark, Μυρμηγκάκι, 137 anonymous edits Examples of groups  Source: http://en.wikipedia.org/w/index.php?oldid=377324328  Contributors: 01001, AxelBoldt, Charles Matthews, Chas zzz brown, Cullinane, Dominus, Doradus, Dysprosia, Eastlaw, Fredrik, Fropuff, JackSchmidt, JeffBobFrank, KathrynLybarger, Lipedia, Michael Hardy, Mikeblas, MithrandirMage, Oleg Alexandrov, PAR, Patrick, Reedy, Seqsea, Thehotelambush, TimothyRias, Toby Bartels, Tosha, 7 anonymous edits Modular representation theory  Source: http://en.wikipedia.org/w/index.php?oldid=405964506  Contributors: Altosax456, Arcfrk, CRGreathouse, Charles Matthews, Davcrav, Dicklyon, Gauge, Geffrey, Hillman, Magmi, Messagetolove, Michael Hardy, R'n'B, R.e.b., Ringspectrum, Rjwilmsi, Silly rabbit, Vanish2, Waltpohl, 63 anonymous edits Conway group  Source: http://en.wikipedia.org/w/index.php?oldid=395314551  Contributors: Charles Matthews, Drschawrz, Geometry guy, Giftlite, JackSchmidt, Jemebius, Kevin Lamoreau, Kwamikagami, Michael Larsen, R.e.b., RFBailey, Radagast3, Schneelocke, Scott Tillinghast, Houston TX, Trovatore, Turgidson, 8 anonymous edits Mathieu group  Source: http://en.wikipedia.org/w/index.php?oldid=387765792  Contributors: BenF, Calcyman, Cullinane, Cyp, Dr. Submillimeter, Drschawrz, EagleFan, Fropuff, FvdP, Geometry guy, Giftlite, Greenfernglade, GregorB, Gro-Tsen, Huppybanny, JackSchmidt, Jim.belk, John Baez, Jtwdog, Keenan Pepper, Marconet, Mboverload, Michael Hardy, Nbarth, R.e.b., RJChapman, Schneelocke, Scott Tillinghast, Houston TX, The Anome, Topbanana, Tosha, Vanish2, WinoWeritas, 22 anonymous edits Sporadic groups  Source: http://en.wikipedia.org/w/index.php?oldid=119491563  Contributors: Almit39, ArnoldReinhold, CRGreathouse, Dominus, Drschawrz, Gaius Cornelius, Geometry guy, Giftlite, Jac16888, JackSchmidt, John of Reading, Kidburla, Michael Hardy, Puffin, R.e.b., Radagast3, Schneelocke, Tobias Bergemann, WinoWeritas, 20 anonymous edits Janko group J1  Source: http://en.wikipedia.org/w/index.php?oldid=370682041  Contributors: Cyp, JackSchmidt, Jemebius, Jim.belk, R.e.b., Remember the dot, Scott Tillinghast, Houston TX, Thomaso, Woohookitty, 1 anonymous edits Janko group J2  Source: http://en.wikipedia.org/w/index.php?oldid=124392814  Contributors: Cyp, DavidCBryant, Giftlite, JackSchmidt, Jim.belk, R.e.b., Scott Tillinghast, Houston TX, 2 anonymous edits Janko group J3  Source: http://en.wikipedia.org/w/index.php?oldid=394978277  Contributors: JackSchmidt, Lexein, R.e.b., Remember the dot, Scott Tillinghast, Houston TX, 2 anonymous edits Janko group J4  Source: http://en.wikipedia.org/w/index.php?oldid=394981293  Contributors: Giftlite, JackSchmidt, Karam.Anthony.K, R.e.b., Remember the dot, Scott Tillinghast, Houston TX, 3 anonymous edits Fischer group  Source: http://en.wikipedia.org/w/index.php?oldid=403610240  Contributors: Carlmckie, Dcoetzee, DroEsperanto, Giftlite, Gro-Tsen, Huppybanny, JackSchmidt, Jiang, Jim.belk, Onebyone, R.e.b., Schneelocke, Scott Tillinghast, Houston TX, 3 anonymous edits Baby Monster group  Source: http://en.wikipedia.org/w/index.php?oldid=398664220  Contributors: Bmonster28, Drschawrz, Farosdaughter, Geometry guy, Gro-Tsen, Huppybanny, Michael Hardy, R.e.b., Schneelocke, 3 anonymous edits Monster group  Source: http://en.wikipedia.org/w/index.php?oldid=405497262  Contributors: 800km3rk, Army1987, AxelBoldt, B.d.mills, BenF, Calabraxthis, Drschawrz, Fredrik, Fuzheado, Gene Ward Smith, Geometry guy, Giftlite, Huppybanny, JackSchmidt, Jemebius, Jitse Niesen, Kevin Lamoreau, Kidburla, Lethe, Linas, Loren Rosen, Michael Hardy, Nbarth, Patrick, Protasis, Qloop, R.e.b., Rjwilmsi, RobHar, Roger Hui, Schneelocke, Schnolle, Scott Tillinghast, Houston TX, Sligocki, Tobias Bergemann, Tomruen, WJBscribe, WinoWeritas, Zundark, 29 anonymous

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