# Group theory

## Group theory Topics

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### Twisted chevalley groups

A finite simple group of Lie-type. The following table summarizes the types of twisted Chevalley groups and their respective orders. In the table, denotes a prime power and the superscript denotes the order of the twisting automorphism.grouporder

### Nilmanifold

Let be a nilpotent, connected, simply connected Lie group, and let be a discrete subgroup of with compact right quotient space. Then is called a nilmanifold.

### Lie group quotient space

The set of left cosets of a subgroup of a topological group forms a topological space. Its topology is defined by the quotient topology from . Namely, the open sets in are the images of the open sets in . Moreover, if is closed, then is a T2-space.

### Solvable lie group

A solvable Lie group is a Lie group which is connected and whose Lie algebra is a solvable Lie algebra. That is, the Lie algebra commutator series(1)eventually vanishes, for some . Since nilpotent Lie algebras are also solvable, any nilpotent Lie group is a solvable Lie group.The basic example is the group of invertible upper triangular matrices with positive determinant, e.g.,(2)such that . The Lie algebra of is its tangent space at the identity matrix, which is the vector space of all upper triangular matrices, and it is a solvable Lie algebra. Its Lie algebra commutator series is given by(3)(4)(5)Any real solvable Lie group is diffeomorphic to Euclidean space. For instance, the group of matrices in the example above is diffeomorphic to , via the Lie group exponential map. However, in general, the exponential map in a solvable Lie algebra need not be surjective...

### Sol geometry

The geometry of the Lie group semidirect product with , where acts on by .

### Lie group

A Lie group is a smooth manifold obeying the group properties and that satisfies the additional condition that the group operations are differentiable.This definition is related to the fifth of Hilbert's problems, which asks if the assumption of differentiability for functions defining a continuous transformation group can be avoided.The simplest examples of Lie groups are one-dimensional. Under addition, the real line is a Lie group. After picking a specific point to be the identity element, the circle is also a Lie group. Another point on the circle at angle from the identity then acts by rotating the circle by the angle In general, a Lie group may have a more complicated group structure, such as the orthogonal group (i.e., the orthogonal matrices), or the general linear group (i.e., the invertible matrices). The Lorentz group is also a Lie group.The tangent space at the identity of a Lie group always has the structure of a Lie algebra, and this..

### Semisimple lie group

A Lie group is called semisimple if its Lie algebra is semisimple. For example, the special linear group and special orthogonal group (over or ) are semisimple, whereas triangular groups are not.

### Nilpotent lie group

A nilpotent Lie group is a Lie group which is connected and whose Lie algebra is a nilpotent Lie algebra . That is, its Lie algebra lower central series(1)eventually vanishes, for some . So a nilpotent Lie group is a special case of a solvable Lie group.The basic example is the group of uppertriangular matrices with 1s on their diagonals, e.g.,(2)which is called the Heisenberg group. Its Lie algebra lower central series is given by(3)(4)(5)Any real nilpotent Lie group is diffeomorphic to Euclidean space. For instance, the group of matrices in the example above is diffeomorphic to , via the Lie group exponential map. In general, the exponential map of a nilpotent Lie algebra is surjective, in contrast to the more general solvable Lie group.

### Nilpotent lie algebra

A Lie algebra is nilpotent when its Lie algebra lower central series vanishes for some . Any nilpotent Lie algebra is also solvable. The basic example of a nilpotent Lie algebra is the vector space of strictly upper triangular matrices, such as the Lie algebra of the Heisenberg group.

### Lie algebra root

The roots of a semisimple Lie algebra are the Lie algebra weights occurring in its adjoint representation. The set of roots form the root system, and are completely determined by . It is possible to choose a set of Lie algebra positive roots, every root is either positive or is positive. The Lie algebra simple roots are the positive roots which cannot be written as a sum of positive roots.The simple roots can be considered as a linearly independent finite subset of Euclidean space, and they generate the root lattice. For example, in the special Lie algebra of two by two matrices with zero matrix trace, has a basis given by the matrices(1)The adjoint representation is given bythe brackets(2)(3)so there are two roots of given by and . The Lie algebraic rank of is one, and it has one positive root...

### Solvable lie algebra

A Lie algebra is solvable when its Lie algebra commutator series, or derived series, vanishes for some . Any nilpotent Lie algebra is solvable. The basic example is the vector space of upper triangular matrices, because every time two such matrices commute, their nonzero entries move further from the diagonal.

### Lie algebra representation

A representation of a Lie algebra is a linear transformationwhere is the set of all linear transformations of a vector space . In particular, if , then is the set of square matrices. The map is required to be a map of Lie algebras so thatfor all . Note that the expression only makes sense as a matrix product in a representation. For example, if and are antisymmetric matrices, then is skew-symmetric, but may not be antisymmetric.The possible irreducible representations of complex Lie algebras are determined by the classification of the semisimple Lie algebras. Any irreducible representation of a complex Lie algebra is the tensor product , where is an irreducible representation of the quotient of the algebra and its Lie algebra radical, and is a one-dimensional representation.A Lie algebra may be associated with a Lie group, in which case it reflects the local structure of the Lie group. Whenever a Lie group has a group representation on , its tangent..

### Cartan subalgebra

Let be a finite-dimensional Lie algebra over some field . A subalgebra of is called a Cartan subalgebra if it is nilpotent and equal to its normalizer, which is the set of those elements such that .It follows from the definition that if is nilpotent, then itself is a Cartan subalgebra of . On the other hand, let be the Lie algebra of all endomorphisms of (for some natural number ), with . Then the set of all endomorphisms of of the form is a Cartan subalgebra of .It can be proved that: 1. If is infinite, then has Cartan subalgebras. 2. If the characteristic of is equal to , then all Cartan subalgebras of have the same dimension. 3. If is algebraically closed and its characteristic is equal to 0, then, given two Cartan subalgebras and of , there is an automorphism of such that . 4. If is semisimple and is an infinite field whose characteristic is equal to 0, then all Cartan subalgebras of are Abelian. Every Cartan subalgebra of a Lie algebra is a maximal nilpotent subalgebra..

### Simple lie algebra

A Lie algebra is said to be simple if it is not Abelian and has no nonzero proper ideals.Over an algebraically closed field of field characteristic 0, every simple Lie algebra is constructed from a simple reduced root system by the Chevalley construction, as described by Humphreys (1977).Over an algebraically closed field of field characteristic , every simple Lie algebra is constructed from a simple reduced root system (as in the characteristic 0 case) or is a Cartan algebra.There also exist simple Lie algebras over algebraically closed fields of field characteristic 2, 3, and 5 that are not constructed from a simple reduced root system and are not Cartan algebras.

### Cartan matrix

A Cartan matrix is a square integer matrix who elements satisfy the following conditions. 1. is an integer, one of . 2. the diagonal entries are all 2. 3. off of the diagonal. 4. iff . 5. There exists a diagonal matrix such that gives a symmetric and positive definite quadratic form. A Cartan matrix can be associated to a semisimple Lie algebra . It is a square matrix, where is the Lie algebra rank of . The Lie algebra simple roots are the basis vectors, and is determined by their inner product, using the Killing form.(1)In fact, it is more a table of values than a matrix. By reordering the basis vectors, one gets another Cartan matrix, but it is considered equivalent to the original Cartan matrix.The Lie algebra can be reconstructed, up to isomorphism, by the generators which satisfy the Chevalley-Serre relations. In fact,(2)where are the Lie subalgebras generated by the generators of the same letter.For example,(3)is a Cartan matrix. The Lie algebra..

### Semisimple lie algebra

A Lie algebra over a field of characteristic zero is called semisimple if its Killing form is nondegenerate. The following properties can be proved equivalent for a finite-dimensional algebra over a field of characteristic 0: 1. is semisimple. 2. has no nonzero Abelian ideal. 3. has zero ideal radical (the radical is the biggest solvable ideal). 4. Every representation of is fully reducible, i.e., is a sum of irreducible representations. 5. is a (finite) direct product of simple Lie algebras (a Lie algebra is called simple if it is not Abelian and has no nonzero ideal ).

### Root system

Let be a Euclidean space, be the dot product, and denote the reflection in the hyperplane bywhereThen a subset of the Euclidean space is called a root system in if: 1. is finite, spans , and does not contain 0, 2. If , the reflection leaves invariant, and 3. If , then . The Lie algebra roots of a semisimple Lie algebra are a root system, in a real subspace of the dual vector space to the Cartan subalgebra. In this case, the reflections generate the Weyl group, which is the symmetry group of the root system.

### Lie algebra lower central series

The lower central series of a Lie algebra is the sequence of subalgebras recursively defined by(1)with . The sequence of subspaces is always decreasing with respect to inclusion or dimension, and becomes stable when is finite dimensional. The notation means the linear span of elements of the form , where and .When the lower central series ends in the zero subspace, the Lie algebra is called nilpotent. For example, consider the Lie algebra of strictly upper triangular matrices, then(2)(3)(4)(5)and . By definition, , where is the term in the Lie algebra commutator series, as can be seen by the example above.In contrast to the nilpotent Lie algebras, the semisimple Lie algebras have a constant lower central series. Others are in between, e.g.,(6)which is semisimple, because the matrix trace satisfies(7)Here, is a general linear Lie algebra and is the special linear Lie algebra...

### Lie algebra commutator series

The commutator series of a Lie algebra , sometimes called the derived series, is the sequence of subalgebras recursively defined by(1)with . The sequence of subspaces is always decreasing with respect to inclusion or dimension, and becomes stable when is finite dimensional. The notation means the linear span of elements of the form , where and .When the commutator series ends in the zero subspace, the Lie algebra is called solvable. For example, consider the Lie algebra of strictly upper triangular matrices, then(2)(3)(4)and . By definition, where is the term in the Lie algebra lower central series, as can be seen by the example above.In contrast to the solvable Lie algebras, the semisimple Lie algebras have a constant commutator series. Others are in between, e.g.,(5)which is semisimple, because the matrix trace satisfies(6)Here, is a general linear Lie algebra and is the special linear Lie algebra...

A Lie algebra is a vector space with a Lie bracket , satisfying the Jacobi identity. Hence any element gives a linear transformation given by(1)which is called the adjoint representation of . It is a Lie algebra representation because of the Jacobi identity,(2)(3)(4)A Lie algebra representation is given by matrices. The simplest Lie algebra is the set of matrices. Consider the adjoint representation of , which has four dimensions and so will be a four-dimensional representation. The matrices(5)(6)(7)(8)give a basis for . Using this basis, the adjoint representation is described by the following matrices:(9)(10)(11)(12)

### Reduced root system

A reduced root system is a root system satisfying the additional property that, if , then the only multiples of in are .

### Lie algebra

A nonassociative algebra obeyed by objects such as the Lie bracket and Poisson bracket. Elements , , and of a Lie algebra satisfy(1)(2)and(3)(the Jacobi identity). The relation implies(4)For characteristic not equal to two, these two relations are equivalent.The binary operation of a Lie algebra is the bracket(5)An associative algebra with associative product can be made into a Lie algebra by the Lie product(6)Every Lie algebra is isomorphic to a subalgebra of some where the associative algebra may be taken to be the linear operators over a vector space (the Poincaré-Birkhoff-Witt theorem; Jacobson 1979, pp. 159-160). If is finite dimensional, then can be taken to be finite dimensional (Ado's theorem for characteristic ; Iwasawa's theorem for characteristic ).The classification of finite dimensional simple Lie algebras over an algebraically closed field of characteristic 0 can be accomplished by (1) determining matrices..

### Automorphism group

The group of functions from an object to itself which preserve the structure of the object, denoted . The automorphism group of a group preserves the multiplication table, the automorphism group of a graph the incidence matrix, and that of a field the addition and multiplication tables.The automorphism group of a graph can be computed in the Wolfram Language using GraphAutomorphismGroup[g].

### Gr&uuml;n's lemma

If is a perfect group, then the group center of the quotient group , where is the group center of , is the trivial group.

### Algebraic loop

A quasigroup with an identity element such that and for any in the quasigroup. All groups are loops.In general, loops are considered to have very little in the way of algebraic structure and it is for that reason that many authors limit their investigation to loops which satisfy various other structural conditions. Common examples of such notions are the left- and right-Bol loop, the Moufang loop (which is both a left-Bol loop and a right-Bol loop simultaneously), and the generalized Bol loop.The above definition of loop is purely algebraic and shouldn't be confused with other notions of loop, such as a closed curves, a multi-component knot or hitch, a graph loop, etc.

### Perfect group

A group that coincides with its commutator subgroup.If is a non-Abelian group, its commutator subgroup is a normal subgroup other than the trivial group. It follows that if is simple, it must be perfect. The converse, however, is not necessarily true. For example, the special linear group is always perfect if (Rose 1994, p. 61), but if is not a power of 2 (i.e., the field characteristic of the finite field is not 2), it is not simple, since its group center contains two elements: the identity matrix and its additive inverse , which are different because .

### Outer automorphism group

A particular type of automorphism group which exists only for groups. For a group , the outer automorphism group is the quotient group , which is the automorphism group of modulo its inner automorphism group.

An additive group is a group where the operation is called addition and is denoted . In an additive group, the identity element is called zero, and the inverse of the element is denoted (minus ). The symbols and terminology are borrowed from the additive groups of numbers: the ring of integers , the field of rational numbers , the field of real numbers , and the field of complex numbers are all additive groups.In general, every ring and every field is an additive group. An important class of examples is given by the polynomial rings with coefficients in a ring . In the additive group of the sum is performed by adding the coefficients of equal terms,(1)Modules, abstractvector spaces, and algebras are all additive groups.The sum of vectors of the vector space is defined componentwise,(2)and so is the sum of matrices with entries in a ring ,(3)which is part of the -module structure of the set of matrices .Any quotient group of an Abelian additive group is again..

### Normal subgroup

Let be a subgroup of a group . The similarity transformation of by a fixed element in not in always gives a subgroup. Iffor every element in , then is said to be a normal subgroup of , written (Arfken 1985, p. 242; Scott 1987, p. 25). Normal subgroups are also known as invariant subgroups or self-conjugate subgroup (Arfken 1985, p. 242).All subgroups of Abelian groups are normal (Arfken1985, p. 242).

### Von dyck's theorem

Let a group have a group presentationso that , where is the free group with basis and is the normal subgroup generated by the . If is a group with and if for all , then there is a surjective homomorphism with for all .

### Frattini extension

If is a group, then the extensions of of order with , where is the Frattini subgroup, are called Frattini extensions.

### Unipotent

A square matrix is said to be unipotent if , where is an identity matrix is a nilpotent matrix (defined by the property that is the zero matrix for some positive integer matrix power . The corresponding identity, for some integer allows this definition to be generalized to other types of algebraic systems.An example of a unipotent matrix is a square matrix whose entries below the diagonal are zero and its entries on the diagonal are one. An explicit example of a unipotent matrix is given byOne feature of a unipotent matrix is that its matrix powers have entries which grow like a polynomial in .A semisimple element of a group is unipotent if is a p-group, where is the generalized fitting subgroup.

### Landau's function

Landau's function is the maximum order of an element in the symmetric group . The value is given by the largest least common multiple of all partitions of the numbers 1 to . The first few values for , 2, ... are 1, 2, 3, 4, 6, 6, 12, 15, 20, 30, ... (OEIS A000793), and have been computed up to by Grantham (1995).Landau showed thatLocal maxima of this function occur at 2, 3, 5, 7, 9, 10, 12, 17, 19, 30, 36, 40,... (OEIS A103635).Let be the greatest prime factor of . Then the first few terms for , 3, ... are 2, 3, 2, 3, 3, 3, 5, 5, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 11, ... (OEIS A129759). Nicolas (1969) showed that . Massias et al. (1988, 1989) showed that for all , , and Grantham (1995) showed that for all , the constant 2.86 may be replaced by 1.328.

### Fourth group isomorphism theorem

The fourth group isomorphism theorem, also called the lattice group isomorphism theorem, lets be a group and let , where indicates that is a normal subgroup of . Then there is a bijection from the set of subgroups of that contain onto the set of subgroups of . In particular, every subgroup is of the form for some subgroup of containing (namely, its preimage in under the natural projection homomorphism from to .) This bijection has the following properties: for all with and , 1. iff 2. If , then 3. , where denotes the subgroup generated by and 4. 5. iff .

### Tightly embedded

is said to be tightly embedded if is odd for all , where is the normalizer of in .

### Lagrange's group theorem

The most general form of Lagrange's group theorem, also known as Lagrange's lemma, states that for a group , a subgroup of , and a subgroup of , , where the products are taken as cardinalities (thus the theorem holds even for infinite groups) and denotes the subgroup index for the subgroup of . A frequently stated corollary (which follows from taking , where is the identity element) is that the order of is equal to the product of the order of and the subgroup index of .The corollary is easily proven in the case of being a finite group, in which case the left cosets of form a partition of , thus giving the order of as the number of blocks in the partition (which is ) multiplied by the number of elements in each partition (which is just the order of ).For a finite group , this corollary gives that the order of must divide the order of . Then, because the order of an element of is the order of the cyclic subgroup generated by , we must have that the order of any element of divides..

### First multiplier theorem

Let be a planar Abelian difference set and be any divisor of . Then is a numerical multiplier of , where a multiplier is defined as an automorphism of a group which takes to a translation of itself for some . If is of the form for relatively prime to the order of , then is called a numerical multiplier.

### Third group isomorphism theorem

Let be a group having normal subgroups and with . Then andwhere indicates that is a normal subgroup of and indicates that and are isomorphic groups.

### First group isomorphism theorem

The first group isomorphism theorem, also known as the fundamental homomorphism theorem, states that if is a group homomorphism, then and , where indicates that is a normal subgroup of , denotes the group kernel, and indicates that and are isomorphic groups.A corollary states that if is a group homomorphism, then 1. is injective iff 2. , where denotes the group order of a group .

### Subgroup index

For a subgroup of a group , the index of , denoted , is the cardinal number of the set of left cosets of in (which is equal to the cardinal number of the set of right cosets of in ).

### Elasticity

For an atomic integral domain (i.e., one in which every nonzero nonunit can be factored as a product of irreducible elements) with the set of irreducible elements, the elasticity of is defined as

### Stabilizer

Let be a permutation group on a set and be an element of . Then(1)is called the stabilizer of and consists of all the permutations of that produce group fixed points in , i.e., that send to itself. For example, the stabilizer of 1 and of 2 under the permutation group is both , and the stabilizer of 3 and of 4 is .More generally, the subset of all images of under permutations of the group (2)is called the group orbit of in .A group's action on an group orbit through is transitive, and so is related to its isotropy group. In particular, the cosets of the isotropy subgroup correspond to the elements in the orbit,(3)where is the orbit of in and is the stabilizer of in . This immediately gives the identity(4)where denotes the order of group (Holton and Sheehan 1993, p. 27)...

### It&ocirc;'s theorem

The dimension of any irreducible representation of a group must be a divisor of the index of each maximal normal Abelian subgroup of .Note that while Itô's theorem was proved by Noboru Itô, Ito'slemma was proven by Kiyoshi Ito.

### Difference set order

Let be group of group order and be a set of elements of . If the set of differences contains every nonzero element of exactly times, then is a -difference set in of order .

### Socle

The socle of a group is the subgroup generated by its minimal normal subgroups. For example, the symmetric group has two nontrivial normal subgroups: and . But contains , so is the only minimal subgroup, and the socle of is .

### Isomorphic groups

Two groups are isomorphic if the correspondence between them is one-to-one and the "multiplication" table is preserved. For example, the point groups and are isomorphic groups, written or (Shanks 1993).

### Difference set

Let be a group of group order and be a set of elements of . If the set of differences contains every nonzero element of exactly times, then is a -difference set in of order . If , the difference set is called planar. The quadratic residues in the finite field form a difference set. If there is a difference set of size in a group , then must be a multiple of , where is a binomial coefficient.Gordon maintains an index of known difference sets.

### Semisimple element

A -element of a group is semisimple if , where is the commuting product of all components of and is the centralizer of .

### Invariant series

An invariant series of a group is a normal seriessuch that each , where means that is a normal subgroup of .

### Semidirect product

A "split" extension of groups and which contains a subgroup isomorphic to with and (Ito 1987, p. 710). Then the semidirect product of a group by a group , denoted (or sometimes ) with homomorphism is given bywhere , , and (Suzuki 1982, p. 67; Scott 1987, p. 213). Note that the semidirect product of two groups is not uniquely defined.The semidirect product of a group by a group can also be defined as a group which is the product of its subgroups and , where is normal in and . If is also normal in , then the semidirect product becomes a group direct product (Shmel'kin 1988, p. 247).

### Identity element

The identity element (also denoted , , or 1) of a group or related mathematical structure is the unique element such that for every element . The symbol "" derives from the German word for unity, "Einheit." An identity element is also called a unit element.

### Crystallography restriction

If a discrete group of displacements in the plane has more than one center of rotation, then the only rotations that can occur are by 2, 3, 4, and 6. This can be shown as follows. It must be true that the sum of the interior angles divided by the number of sides is a divisor of .where is an integer. Therefore, symmetry will be possible only forwhere is an integer. This will hold for 1-, 2-, 3-, 4-, and 6-fold symmetry. That it does not hold for is seen by noting that corresponds to . The case requires that (impossible), and the case requires that (also impossible).The point groups that satisfy the crystallographic restriction are called crystallographic point groups.Although -fold rotations for differing from 2, 3, 4, and 6 are forbidden in the strict sense of perfect crystallographic symmetry, there are exotic materials called quasicrystals that display these symmetries. In 1984, D. Shechtman discovered a class of aluminum alloys whose X-ray..

### Second group isomorphism theorem

The second, or diamond, group isomorphism theorem, states that if is a group with , and , then and , where indicates that is a normal subgroup of and indicates that and are isomorphic groups.This theorem is so named because of the diamond shaped lattice of subgroups of involved.

### Homogeneous permutation group

A permutation group is -homogeneous if it is transitive on unordered k-subsets of .The projective special linear group is 3-homogeneous if .

### Cross number

The cross number of a zero-system of is defined asThe cross number of a group has two different definitions. 1. Anderson and Chapman (2000) define the cross number of as . 2. Chapman (1997) defines , where . A value of the cross number: for a prime and , . A stronger statement is that any finite Abelian group is cyclic of prime power order iff .

### Schur's representation lemma

If on and on are irreducible representations and is a linear map such that for all and group , then or is invertible. Furthermore, if in a vector space over complex numbers, then is a scalar.

### Coset

For a subgroup of a group and an element of , define to be the set and to be the set . A subset of of the form for some is said to be a left coset of and a subset of the form is said to be a right coset of .For any subgroup , we can define an equivalence relation by if for some . The equivalence classes of this equivalence relation are exactly the left cosets of , and an element of is in the equivalence class . Thus the left cosets of form a partition of .It is also true that any two left cosets of have the same cardinal number, and in particular, every coset of has the same cardinal number as , where is the identity element. Thus, the cardinal number of any left coset of has cardinal number the order of .The same results are true of the right cosets of and, in fact, one can prove that the set of left cosets of has the same cardinal number as the set of right cosets of ...

### Right transversal

Let be a subgroup of . A subset of elements of is called a right transversal of if contains exactly one element of each right coset of .

### Group torsion

If is a group, then the torsion elements of (also called the torsion of ) are defined to be the set of elements in such that for some natural number , where is the identity element of the group .In the case that is Abelian, is a subgroup and is called the torsion subgroup of . If consists only of the identity element, the group is called torsion-free.

### Right coset

Consider a countable subgroup with elements and an element not in , then for , 2, ... constitute the right coset of the subgroup with respect to .

### Conjugate element

Given a group with elements and , there must be an element which is a similarity transformation of so and are conjugate with respect to . Conjugate elements have the following properties: 1. Every element is conjugate with itself. 2. If is conjugate with with respect to , then is conjugate to with respect to . 3. If is conjugate with and , then and are conjugate with each other.

### Rearrangement theorem

Each row and each column in the group multiplication table lists each of the group elements once and only once. From this, it follows that no two elements may be in the identical location in two rows or two columns. Thus, each row and each column is a rearranged list of the group elements. Stated otherwise, given a group of distinct elements , the set of products reproduces the original distinct elements in a new order.

### Group rank

For any prime number and any positive integer , the -rank of a finitely generated Abelian group is the number of copies of the cyclic group appearing in the Kronecker decomposition of (Schenkman 1965). The free (or torsion-free) rank of is the number of copies of appearing in the same decomposition. It can be characterized as the maximal number of elements of which are linearly independent over . Since it is also equal to the dimension of as a vector space over , it is often called the rational rank of . Munkres (1984) calls it the Betti number of .Most authors refer to simply as the "rank" of (Kargapolov and Merzljakov 1979), whereas others (Griffith 1970) use the word "rank" to denote the sum . In this latter meaning, the rank of is the number of direct summands appearing in the Kronecker decomposition of ...

### Quasithin theorem

In the classical quasithin case of the quasithin theorem, if a group does not have a "strongly embedded" subgroup, then is a group of Lie-type in characteristic 2 of Lie rank 2 generated by a pair of parabolic subgroups and , or is one of a short list of exceptions.

### Conjugacy class

A complete set of mutually conjugate group elements. Each element in a group belongs to exactly one class, and the identity element () is always in its own class. The conjugacy class orders of all classes must be integral factors of the group order of the group. From the last two statements, a group of prime order has one class for each element. More generally, in an Abelian group, each element is in a conjugacy class by itself.Two operations belong to the same class when one may be replaced by the other in a new coordinate system which is accessible by a symmetry operation (Cotton 1990, p. 52). These sets correspond directly to the sets of equivalent operations.To see how to compute conjugacy classes, consider the dihedral group D3, which has the following multiplication table.11111111 is always in a conjugacy class of its own. To find another conjugacy class take some element, say , and find the results of all similarity transformations on . For..

### Primary representation

Let be a unitary representation of a group on a separable Hilbert space, and let be the smallest weakly closed algebra of bounded linear operators containing all for . Then is primary if the center of consists of only scalar operations.

### Group orthogonality theorem

Let be a representation for a group of group order , thenThe proof is nontrivial and may be found in Eyring et al. (1944).

### Composition series

Every finite group of order greater than one possesses a finite series of subgroups, called a composition series, such thatwhere is a maximal subgroup of and means that is a normal subgroup of . A composition series is therefore a normal series without repetition whose factors are all simple (Scott 1987, p. 36).The quotient groups , , ..., , are called composition quotient groups.

### Group orbit

In celestial mechanics, the fixed path a planet traces as it moves around the sun is called an orbit. When a group acts on a set (this process is called a group action), it permutes the elements of . Any particular element moves around in a fixed path which is called its orbit. In the notation of set theory, the group orbit of a group element can be defined as(1)where runs over all elements of the group . For example, for the permutation group , the orbits of 1 and 2 are and the orbits of 3 and 4 are .A group fixed point is an orbit consisting of a single element, i.e., an element that is sent to itself under all elements of the group. The stabilizer of an element consists of all the permutations of that produce group fixed points in , i.e., that send to itself. The stabilizers of 1 and 2 under are therefore , and the stabilizers of 3 and 4 are .Note that if then , because iff . Consequently, the orbits partition and, given a permutation group on a set , the orbit of an element is..

### Orthogonal group representations

Two representations of a group and are said to be orthogonal iffor , where the sum is over all elements of the representation.

### Centralizer

The centralizer of an element of a group is the set of elements of which commute with ,Likewise, the centralizer of a subgroup of a group is the set of elements of which commute with every element of ,The centralizer always contains the group center of the group and is contained in the corresponding normalizer. In an Abelian group, the centralizer is the whole group.

### Normalizer

The set of elements of a group such thatis said to be the normalizer with respect to a subset of group elements . If is a subgroup of , is also a subgroup containing .

### Group generators

A set of generators is a set of group elements such that possibly repeated application of the generators on themselves and each other is capable of producing all the elements in the group. Cyclic groups can be generated as powers of a single generator. Two elements of a dihedral group that do not have the same sign of ordering are generators for the entire group.The Cayley graph of a group and a subset of elements (excluding the identity element) is connected iff the subset generates the group.

### Normal series

A normal series of a group is a finite sequence of normal subgroups such that

### Group fixed point

The set of points of fixed by a group action are called the group's set of fixed points, defined byIn some cases, there may not be a group action, but a single operator . Then still makes sense even when is not invertible (as is the case in a group action).

### Normal factor

Let be a group with normal series (, , ..., ). A normal factor of is a quotient group for some index . is a solvable group iff all normal factors are Abelian.

### Group cycle

A cycle of a finite group is a minimal set of elements such that , where is the identity element. A diagram of a group showing every cycle in the group is known as a cycle graph (Shanks 1993, p. 83).For example, the modulo multiplication group (i.e., the group of residue classes relatively prime to 5 under multiplication mod 5) has elements and cycles , , , and . The corresponding cycle graph is illustrated above.

### Burnside's conjecture

In Note M, Burnside (1955) states, "The contrast that these results shew between groups of odd and of even order suggests inevitably that simple groups of odd order do not exist." Of course, simple groups of prime order do exist, namely the groups for any prime . Therefore, Burnside conjectured that every finite simple group of non-prime order must have even order. The conjecture was proven true by Feit and Thompson (1963).

### Group convolution

The convolution of two complex-valued functions on a group is defined aswhere the support (set which is not zero) of each functionis finite.

### Group character

The group theoretical term for what is known to physicists, by way of its connection with matrix traces, as the trace. The powerful group orthogonality theorem gives a number of important properties about the structures of groups, many of which are most easily expressed in terms of characters. In essence, group characters can be thought of as the matrix traces of a special set of matrices (a so-called irreducible representation) used to represent group elements and whose multiplication corresponds to the multiplication table of the group.All members of the same conjugacy class in the same representation have the same character. Members of other conjugacy classes may also have the same character, however. An (abstract) group can be identified by a listing of the characters of its various representations, known as a character table. However, there exist nonisomorphic groups which nevertheless have the same character table, for example (the..

### Multiplicative inverse

In a monoid or multiplicative group where the operation is a product , the multiplicative inverse of any element is the element such that , with 1 the identity element.The multiplicative inverse of a nonzero number is its reciprocal (zero is not invertible). For complex ,The inverse of a nonzero real quaternion (where are real numbers, and not all of them are zero) is its reciprocalwhere .The multiplicative inverse of a nonsingular matrixis its matrix inverse.To detect the multiplicative inverse of a given element in the multiplication table of finite multiplicative group, traverse the element's row until the identity element 1 is encountered, and then go up to the top row. In this way, it can be immediately determined that is the multiplicative inverse of in the multiplicative group formed by all complex fourth roots of unity...

### Aschbacher's component theorem

Suppose that (the commuting product of all components of ) is simple and contains a semisimple group involution. Then there is some semisimple group involution such that has a normal subgroup which is either quasisimple or isomorphic to and such that is tightly embedded.

### Global c(g;t) theorem

If a Sylow 2-subgroup of lies in a unique maximal 2-local of , then is a "strongly embedded" subgroup of , and is known.

### Antihomomorphism

If a map from a group to a group satisfies for all , then is said to be an antihomomorphism.

### M&ouml;bius problem

Let be a free Abelian semigroup, where is the identity element, and let be the Möbius function. Define on the elements of the semigroup analogously to the definition of (as if is the product of distinct primes) by regarding generators of the semigroup as primes. Then the Möbius problem asks if the properties 1. implies for , where has the linear order , 2. for all , imply thatfor all . Informally, the problem asks "Is the multiplication law on the positive integers uniquely determined by the values of the Möbius function and the property that multiplication respects order?The problem is known to be true for all if for all (Flath and Zulauf 1995).

### Antiautomorphism

If a map from a group to a group satisfies for all , then is said to be an antihomomorphism. Moreover, if and are isomorphic, then is said to be an antiautomorphism.

### Maximal tori theorem

Let be a maximal torus of a group , then intersects every conjugacy class of , i.e., every element is conjugate to a suitable element in . The theorem is due to É. Cartan.

In an additive group , the additive inverse of an element is the element such that , where 0 is the additive identity of . Usually, the additive inverse of is denoted , as in the additive group of integers , of rationals , of real numbers , and of complex numbers , where The same notation with the minus sign is used to denote the additive inverse of a vector,(1)of a polynomial,(2)of a matrix(3)and, in general, of any element in an abstractvector space or a module.

The identity element of an additive group , usually denoted 0. In the additive group of vectors, the additive identity is the zero vector , in the additive group of polynomials it is the zero polynomial , in the additive group of matrices it is the zero matrix.

### Left transversal

Let be a subgroup of . A subset of elements of is called a left transversal of if contains exactly one element of each left coset of .

### Abstract group

An abstract group is a group characterized only by its abstract properties and not by the particular representations chosen for elements. For example, there are two distinct abstract groups on four elements: the vierergruppe and the cyclic group C4. A number of particular examples of the abstract group are the point groups (unfortunately, the symbols for the point groups are the same as those for the abstract cyclic groups to which they are isomorphic) and .

### Left coset

For a group , consider a subgroup with elements and an element of not in , then for , 2, ... constitute the left coset of the subgroup with respect to .

### Inner automorphism

An inner automorphism of a group is an automorphism of the form , where is a fixed element of .The automorphism of the symmetric group that maps the permutation to is an inner automorphism, since .

### Transitive group action

A group action is transitive if it possesses only a single group orbit, i.e., for every pair of elements and , there is a group element such that . In this case, is isomorphic to the left cosets of the isotropy group, . The space , which has a transitive group action, is called a homogeneous space when the group is a Lie group.If, for every two pairs of points and , there is a group element such that , then the group action is called doubly transitive. Similarly, a group action can be triply transitive and, in general, a group action is -transitive if every set of distinct elements has a group element such that .

### Mulliken symbols

Symbols used to identify irreducible representations of groups: singly degenerate state which is symmetric with respect to rotation about the principal axis, singly degenerate state which is antisymmetric with respect to rotation about the principal axis, doubly degenerate, triply degenerate, (gerade, symmetric) the sign of the wavefunction does not change on inversion through the center of the atom, (ungerade, antisymmetric) the sign of the wavefunction changes on inversion through the center of the atom, (on or ) the sign of the wavefunction does not change upon rotation about the center of the atom, (on or ) the sign of the wavefunction changes upon rotation about the center of the atom, ' = symmetric with respect to a horizontal symmetry plane , " = antisymmetric with respect to a horizontal symmetry plane . ..

### Quotient group

For a group and a normal subgroup of , the quotient group of in , written and read " modulo ", is the set of cosets of in . Quotient groups are also called factor groups. The elements of are written and form a group under the normal operation on the group on the coefficient . Thus,Since all elements of will appear in exactly one coset of the normal subgroup , it follows thatwhere denotes the order of a group. This is also a consequence of Lagrange's group theorem with and Although the slash notation conflicts with that for an extension field, the meaning can be determined based on context.

### Isoclinic groups

Two groups and are said to be isoclinic if there are isomorphisms and , where is the group center of the group, which identify the two commutator maps.

### Inner automorphism group

A particular type of automorphism group which exists only for groups. For a group , the inner automorphism group is defined bywhere is an automorphism of defined by

### Bimonster

The wreathed product of the monster group by . The bimonster is a quotient of the Coxeter group with the above Coxeter-Dynkin diagram. This had been conjectured by Conway, but was proven around 1990 by Ivanov and Norton. If the parameters in Coxeter's notation are written side by side, the bimonster can be denoted by the beast number 666.

### Proper subgroup

A proper subgroup is a proper subset of group elements of a group that satisfies the four group requirements. " is a proper subgroup of " is written . The group order of any subgroup of a group of group order must be a divisor of .

### Orthomorphism

Let be a group and n permutation of . Then is an orthomorphism of if the self-mapping of defined by is also an permutation of .

### Semigroup

A mathematical object defined for a set and a binary operator in which the multiplication operation is associative. No other restrictions are placed on a semigroup; thus a semigroup need not have an identity element and its elements need not have inverses within the semigroup. A semigroup is an associative groupoid. A semigroup with an identity is called a monoid.A semigroup can be empty. The numbers of nonisomorphic semigroups of orders 1, 2,... are 1, 5, 24, 188, 1915, ... (OEIS A027851).The number of semigroups of order , 2, ... with one idempotent are 1, 2, 5, 19, 132, 3107, 623615, ... (OEIS A002786), and with two idempotents are 2, 7, 37, 216, 1780, 32652, ... (OEIS A002787). The number of semigroups having , 3, ... idempotents are 1, 2, 6, 26, 135, 875, ... (OEIS A002788).

### Quasigroup

A groupoid such that for all , there exist unique such that(1)(2)No other restrictions are applied; thus a quasigroup need not have an identity element, not be associative, etc. Quasigroups are precisely groupoids whose multiplication tables are Latin squares. A quasigroup can be empty.

### Groupoid

There are at least three definitions of "groupoid" currently in use.The first type of groupoid is an algebraic structure on a set with a binary operator. The only restriction on the operator is closure (i.e., applying the binary operator to two elements of a given set returns a value which is itself a member of ). Associativity, commutativity, etc., are not required (Rosenfeld 1968, pp. 88-103). A groupoid can be empty. The numbers of nonisomorphic groupoids of this type having , 2, ... elements are 1, 10, 3330, 178981952, ... (OEIS A001329), and the corresponding numbers of nonisomorphic and nonantiisomorphic groupoids are 1, 7, 1734, 89521056, ... (OEIS A001424). An associative groupoid is called a semigroup.The second type of groupoid is, roughly, a category which is "group-like" in the sense that every morphism (or arrow) is invertible. To make this notion more precise, one says that a groupoid is a category..

### Generalized bol loop

A algebraic loop is a generalized Bol loop if for all elements , , and of ,for some map . As the name suggests, these are generalizations of Bol loops; in particular, a Bol loop is a generalized Bol loop with respect to the identity map .One can show that there is an algebraic duality between generalized Bol loops and algebraic loops which satisfy the half-Bol identity (Adeniran and Solarin 1999).

### Zero group

The singleton set , with respect to the trivial group structure defined by the addition . The element 0 is the additive identity element of the group, and also the additive inverse of itself.The zero group is a minimal example of group, hence it is called a trivial group. Another example of trivial group is the multiplicative group , where .

### Nilpotent group

A group is nilpotent if the upper central sequenceof the group terminates with for some .Nilpotent groups have the property that each proper subgroup is properly contained in its normalizer. A finite nilpotent group is the direct product of its Sylow p-subgroups.

There are at least two distinct notions known as the Whitehead group.Given an associative ring with unit, the Whitehead group associated to is the commutative quotient group(1)where is the union over all natural numbers of the general linear groups and where is the normal subgroup generated by all elementary matrices.Note that the commutativity of stems from the fact (proven by Whitehead) that is the commutator subgroup of .The second definition, though different, is related to the first. Given a multiplicative group with integral group ring , there exist natural homomorphisms(2)In this context, one can define the Whitehead group as the cokernel(3)

### Free abelian group

A free Abelian group is a group with a subset which generates the group with the only relation being . That is, it has no group torsion. All such groups are a direct product of the integers , and have rank given by the number of copies of . For example, is a free Abelian group of rank 2. A minimal subset , ..., that generates a free Abelian group is called a basis, and gives asA free Abelian group is an Abelian group, but is not a free group (except when it has rank one, i.e., ). Free Abelian groups are the free modules in the case when the ring is the ring of integers .

### Multiplicative group

A group whose group operation is identified with multiplication. As with normal multiplication, the multiplication operation on group elements is either denoted by a raised dot or omitted entirely, giving the notation or . In a multiplicative group, the identity element is denoted 1, and the inverse of the element is written as , voiced " inverse." This notation and terminology is borrowed from the multiplicative groups formed by numbers, where the operation is the usual arithmetical product, the identity element is the number 1, and the inverse coincides with the multiplicative reciprocal.The simplest examples are the trivial group and , the latter of which is isomorphic to the cyclic additive group . The elements of are the square roots of unity, and in general, the set of all complex th roots of unity is a cyclic multiplicative group of order ,(1)where the generator is any primitive th root of unity. These groups are all subgroups..

### Monodromy group

A technically defined group characterizing a system oflinear differential equationsfor , ..., , where are complex analytic functions of in a given complex domain.

### Fitting subgroup

The fitting subgroup is the subgroup generated by all normal nilpotent subgroups of a group , denoted .In the case of a finite group, the subgroup generated will itself be a normal nilpotent subgroup, and hence the unique largest normal nilpotent subgroup.The generalized fitting subgroup is defined by , where is the commuting product of all components of and is the fitting subgroup of .

### Transitive group

Transitivity is a result of the symmetry in the group. A group is called transitive if its group action (understood to be a subgroup of a permutation group on a set ) is transitive. In other words, if the group orbit is equal to the entire set for some element , then is transitive.A group is called k-transitive if there exists a set of elements on which the group acts faithfully and -transitively. It should be noted that transitivity computed from a particular permutation representation may not be the (maximal) transitivity of the abstract group. For example, the Higman-Sims group has both a 2-transitive representation of degree 176, and a 1-transitive representation of degree 100. Note also that while -transitivity of groups is related to -transitivity of graphs, they are not identical concepts.The symmetric group is -transitive and the alternating group is -transitive. However, multiply transitive finite groups are rare. In fact, they have..

### Modular group gamma_0

Let be a positive integer, then is defined as the set of all matrices in the modular group Gamma with . is a subgroup of . For any prime , the setis a fundamental region of the subgroup , where and (Apostol 1997).

### Elliptic group modulo p

denotes the elliptic group modulo whose elements are 1 and together with the pairs of integers with satisfying(1)with and integers such that(2)Given , define(3)The group order of is given by(4)where is the Legendre symbol, although this formula quickly becomes impractical. However, it has been proven that(5)Furthermore, for a prime and integer in the above interval, there exists and such that(6)and the orders of elliptic groups mod are nearly uniformly distributed in the interval.

### M&ouml;bius group

The equationrepresents an -dimensional hypersphere as a quadratic hypersurface in an -dimensional real projective space , where are homogeneous coordinates in . Then the group of projective transformations which leave invariant is called the Möbius group.

### Subgroup

A subgroup is a subset of group elements of a group that satisfies the four group requirements. It must therefore contain the identity element. " is a subgroup of " is written , or sometimes (e.g., Scott 1987, p. 16).The order of any subgroup of a group of order must be a divisor of .A subgroup of a group that does not include the entire group itself is known as a proper subgroup, denoted or .

### Metacyclic group

There are two definitions of a metacyclic group. 1. A metacyclic group is a group such that both its commutator subgroup and the quotient group are cyclic (Rose 1994, p. 247). 2. A group is metacyclic if it has a cyclic normal subgroup such that the quotient group is also cyclic (Rose 1994, p. 56). In general, a group may be metacyclic according to the second definition and fail the first one. For example, the quaternion group has a normal cyclic subgroup of order 4, thus it satisfies definition (2). On the other hand, the commutant consists of two elements , the quotient is isomorphic to the finite group C2×C2, and thus the group is not cyclic.The first definition is more classical, but nowadays essentially all algebraists use the second definition, which is the one used in the remainder of this article.Metacyclic groups are solvable and have a compositionseries of length two.A complete classification of finite metacyclic groups..

### Solvable group

A solvable group is a group having a normal series such that each normal factor is Abelian. The special case of a solvable finite group is a group whose composition indices are all prime numbers. Solvable groups are sometimes called "soluble groups," a turn of phrase that is a source of possible amusement to chemists.The term "solvable" derives from this type of group's relationship to Galois's theorem, namely that the symmetric group is unsolvable for while it is solvable for , 2, 3, and 4. As a result, the polynomial equations of degree are (in general) not solvable using finite additions, multiplications, divisions, and root extractions.A major building block for the classification of finite simple groups was the Feit-Thompson theorem, which proved that every group of odd order is solvable. This proof took up an entire journal issue.Every finite group of order , every Abelian group, and every subgroup of a solvable group..

### Metabelian group

A group such that the quotient group , where is the group center of , is Abelian. An equivalent condition is that the commutator subgroup is contained in .

### Conjugate subgroup

A subgroup of an original group has elements . Let be a fixed element of the original group which is not a member of . Then the transformation , (, 2, ...) generates the so-called conjugate subgroup . If, for all , , then is a normal (also called "self-conjugate" or "invariant") subgroup.All subgroups of an Abeliangroup are normal.

### Lorentz group

The Lorentz group is the group of time-preserving linear isometries of Minkowski space with the Minkowski metric(where the convention is used). It is also the group of isometries of three-dimensional hyperbolic geometry. It is time-preserving in the sense that the unit time vector is sent to another vector such that .A consequence of the definition of the Lorentz group is that the full group of time-preserving isometries of Minkowski space is the group direct product of the group of translations of (i.e., itself, with addition as the group operation), with the Lorentz group, and that the full isometry group of the Minkowski is a group extension of by the product .The Lorentz group is invariant under space rotations and Lorentztransformations.

### Commutator subgroup

The commutator subgroup (also called a derived group) of a group is the subgroup generated by the commutators of its elements, and is commonly denoted or . It is the unique smallest normal subgroup of such that is Abelian (Rose 1994, p. 59). It can range from the identity subgroup (in the case of an Abelian group) to the whole group. Note that not every element of the commutator subgroup is necessarily a commutator.For instance, in the quaternion group (, , , ) with eight elements, the commutators form the subgroup . The commutator subgroup of the symmetric group is the alternating group. The commutator subgroup of the alternating group is the whole group . When , is a simple group and its only nontrivial normal subgroup is itself. Since is a nontrivial normal subgroup, it must be .The first homology of a group is the Abelianization..

### Kummer group

A group of linear fractional transformations which transform the arguments of Kummer solutions to the hypergeometric differential equation into each other. Define(1)(2)then the elements of the group are .

The quotient spaceof the Whitehead group is known as the reduced Whitehead group. Here, the element denotes the order-2 element corresponding to the unit where is the collection of invertible matrices with coefficients in an associative ring with unit, i.e., the collection of all units in an associative unit ring .

### Isotropy group

Some elements of a group acting on a space may fix a point . These group elements form a subgroup called the isotropy group, defined byFor example, consider the group of all rotations of a sphere . Let be the north pole . Then a rotation which does not change must turn about the usual axis, leaving the north pole and the south pole fixed. These rotations correspond to the action of the circle group on the equator.When two points and are on the same group orbit, say , then the isotropy groups are conjugate subgroups. More precisely, . In fact, any subgroup conjugate to occurs as an isotropy group to some point on the same orbit as .

### Picard group

Let be a number field and let be an order in . Then the set of equivalence classes of invertible fractional ideals of forms a multiplicative Abelian group called the Picard group of .If is a maximal order, i.e., the ring of integers of , then every fractional ideal of is invertible and the Picard group of is the class group of . The order of the Picard group of is sometimes called the class number of . If is maximal, then the order of the Picard group is equal to the class number of .

### Symplectic group

For every even dimension , the symplectic group is the group of matrices which preserve a nondegenerate antisymmetric bilinear form , i.e., a symplectic form.Every symplectic form can be put into a canonical form by finding a symplectic basis. So, up to conjugation, there is only one symplectic group, in contrast to the orthogonal group which preserves a nondegenerate symmetric bilinear form. As with the orthogonal group, the columns of a symplectic matrix form a symplectic basis.Since is a volume form, the symplectic group preserves volume and vector space orientation. Hence, . In fact, is just the group of matrices with determinant 1. The three symplectic (0,1)-matrices are therefore(1)The matrices(2)and(3)are in , where(4)In fact, both of these examples are 1-parameter subgroups.A matrix can be tested to see if it is symplectic using the WolframLanguage code: SymplecticForm[n_Integer] := Join[PadLeft[IdentityMatrix[n], {n,..

### Action

Let denote the group of all invertible maps and let be any group. A homomorphism is called an action of on . Therefore, satisfies 1. For each , is a map . 2. . 3. , where is the group identity in . 4. .

### Short exact sequence

A short exact sequence of groups , , and is given by two maps and and is written(1)Because it is an exact sequence, is injective, and is surjective. Moreover, the group kernel of is the image of . Hence, the group can be considered as a (normal) subgroup of , and is isomorphic to .A short exact sequence is said to split if there is a map such that is the identity on . This only happens when is the direct product of and .The notion of a short exact sequence also makes sense for modules and sheaves. Given a module over a unit ring , all short exact sequences(2)are split iff is projective, and all short exact sequences(3)are split iff is injective.A short exact sequence of vector spaces is alwayssplit.

### Jacobi identities

"The" Jacobi identity is a relationship(1)between three elements , , and , where is the commutator. The elements of a Lie algebra satisfy this identity.Relationships between the Q-functions are also known as Jacobi identities:(2)equivalent to the Jacobi triple product (Borweinand Borwein 1987, p. 65) and(3)where(4) is the complete elliptic integral of the first kind, and . Using Weber functions(5)(6)(7)(5) and (6) become(8)(9)(Borwein and Borwein 1987, p. 69).

### Irreducible representation

An irreducible representation of a group is a group representation that has no nontrivial invariant subspaces. For example, the orthogonal group has an irreducible representation on .Any representation of a finite or semisimple Lie group breaks up into a direct sum of irreducible representations. But in general, this is not the case, e.g., has a representation on by(1)i.e., . But the subspace is fixed, hence is not irreducible, but there is no complementary invariant subspace.The irreducible representation has a number of remarkable properties, as formalized in the group orthogonality theorem. Let the group order of a group be , and the dimension of the th representation (the order of each constituent matrix) be (a positive integer). Let any operation be denoted , and let the th row and th column of the matrix corresponding to a matrix in the th irreducible representation be . The following properties can be derived from the group orthogonality..

### Induced representation

If a subgroup of has a group representation , then there is a unique induced representation of on a vector space . The original space is contained in , and in fact,(1)where is a copy of . The induced representation on is denoted .Alternatively, the induced representation is the CG-module(2)Also, it can be viewed as -valued functions on which commute with the action.(3)The induced representation is also determined by its universalproperty:(4)where is any representation of . Also, the induced representation satisfies the following formulas. 1. . 2. for any group representation . 3. when . Some of the group characters of can be calculated from the group characters of , as induced representations, using Frobenius reciprocity. Artin's reciprocity theorem says that the induced representations of cyclic subgroups of a finite group generates a lattice of finite index in the lattice of virtual characters. Brauer's theorem says that the virtual characters..

### Group representation restriction

A group representation of a group on a vector space can be restricted to a subgroup . For example, the symmetric group on three letters has a representation on by(1)(2)(3)(4)(5)(6)that can be restricted to the subgroup of group order3,(7)(8)(9)

### Group representation

A representation of a group is a group action of on a vector space by invertible linear maps. For example, the group of two elements has a representation by and . A representation is a group homomorphism .Most groups have many different representations, possibly on different vector spaces. For example, the symmetric group has a representation on by(1)where is the permutation symbol of the permutation . It also has a representation on by(2)A representation gives a matrix for each element, and so another representation of is given by the matrices(3)Two representations are considered equivalent if they are similar. For example, performing similarity transformations of the above matrices by(4)gives the following equivalent representation of ,(5)Any representation of can be restricted to a representation of any subgroup , in which case, it is denoted . More surprisingly, any representation on can be extended to a representation of , on a larger..

### Faithful representation

A representation of a group is faithful if it is one-to-one, i.e., if implies for . Equivalently, is faithful if implies , where is the dimension of , is the identity matrix, and is the identity element of .

### Special unitary group

The special unitary group is the set of unitary matrices with determinant (having independent parameters). is homeomorphic with the orthogonal group . It is also called the unitary unimodular group and is a Lie group.Special unitary groups can be represented by matrices(1)where and are the Cayley-Klein parameters. The special unitary group may also be represented by matrices(2)or the matrices(3)(4)(5)The order representation is(6)The summation is terminated by putting . The group character is given by(7)(8)

### Special orthogonal group

The special orthogonal group is the subgroup of the elements of general orthogonal group with determinant 1. (often written ) is the rotation group for three-dimensional space.

### Orthogonal group

For every dimension , the orthogonal group is the group of orthogonal matrices. These matrices form a group because they are closed under multiplication and taking inverses.Thinking of a matrix as given by coordinate functions, the set of matrices is identified with . The orthogonal matrices are the solutions to the equations(1)where is the identity matrix, which are redundant. Only of these are independent, leaving "free variables." In fact, the orthogonal group is a smooth -dimensional submanifold.Because the orthogonal group is a group and a manifold, it is a Lie group. has a submanifold tangent space at the identity that is the Lie algebra of antisymmetric matrices . In fact, the orthogonal group is a compact Lie group.The determinant of an orthogonal matrix is either 1 or , and so the orthogonal group has two components. The component containing the identity is the special orthogonal group . For example, the group has group action..

### Maschke's theorem

If a matrix group is reducible, then it is completely reducible, i.e., if the matrix group is equivalent to the matrix group in which every matrix has the reduced formthen it is equivalent to the matrix group obtained by putting .

### Heisenberg group

The Heisenberg group in complex variables is the group of all with and having multiplication(1)where is the adjoint. The Heisenberg group is isomorphic to the group of matrices(2)and satisfies(3)Every finite-dimensional unitary representation is trivial on and therefore factors to a group representation of the quotient .

### Moufang loop

An algebraic loop is a Moufang loop if all triples of elements , , and in satisfy the Moufang identities, i.e., if 1. , 2. , 3. , and 4. . One can show that an algebraic loop which satisfies both the left and right Bol identities is Moufang.

### Free idempotent monoid

A free idempotent monoid is a monoid that satisfies the identity and is generated by a set of elements. If the generating set of such a monoid is finite, then so is the free idempotent monoid itself. The number of elements in the monoid depends on the size of the generating set, and the size the generating set uniquely determines a free idempotent monoid. On zero letters, the free idempotent monoid has one element (the identity). With one letter, the free idempotent monoid has two elements . With two letters, it has seven elements: . In general, the numbers of elements in the free idempotent monoids on letters are 1, 2, 7, 160, 332381, ... (OEIS A005345). These are given by the analytic expressionwhere is a binomial coefficient. The product can be done analytically, giving the sumin terms of derivatives of the polylogarithm with respect to its index and the Lerch transcendent with respect to its second argument...

### Monoid

A monoid is a set that is closed under an associative binary operation and has an identity element such that for all , . Note that unlike a group, its elements need not have inverses. It can also be thought of as a semigroup with an identity element.A monoid must contain at least one element.A monoid that is commutative is, not surprisingly, known as a commutativemonoid.

### Commutative monoid

A monoid that is commutative i.e., a monoid such that for every two elements and in , . This means that commutative monoids are commutative, associative, and have an identity element.For example, the nonnegative integers under addition form a commutative monoid. The integers under the operation with all form a commutative monoid. This monoid collapses to a group only if and are restricted to the integers 0, 1, ..., , since only then do the elements have unique additive inverses. Similarly, the integers under the operation also forms a commutative monoid.The numbers of commutative monoids of orders , 2, ... are 1, 2, 5, 19, 78, 421, 2637, ... (OEIS A058131).

### Topological groupoid

A topological groupoid over is a groupoid such that and are topological spaces and , and multiplication are continuous maps. Here, and are maps from onto .

### Lie groupoid

A Lie groupoid over is a groupoid for which and are differentiable manifolds and and multiplication are differentiable maps. Furthermore, the derivatives of and are required to have maximal matrix rank everywhere. Here, and are maps from onto .

### Submonoid

A submonoid is a subset of the elements of a monoid that are themselves a monoid under the same monoid operation. For example, consider the monoid formed by the nonnegative integers under the operation . Then restricting and from all the integers to the set of elements forms a submonoid of the monoid under the operation .

### Inverse semigroup

A semigroup is said to be an inverse semigroup if, for every in , there is a unique (called the inverse of ) such that and . This is equivalent to the condition that every element has at least one inverse and that the idempotents of commute (Lawson 1999). Note that if is an inverse of , then is an idempotent.

### Braun's conjecture

Let be an infinite Abelian semigroup with linear order such that is the unit element and implies for . Define a Möbius function on by andfor , 3, .... Further suppose that (the true Möbius function) for all . Then Braun's conjecture states thatfor all .

### Bol loop

The term Bol loop refers to either of two classes of algebraic loops satisfying the so-called Bol identities. In particular, a left Bol loop is an algebraic loop which, for all , , and in , satisfies the left Bol relationSimilarly, is a right Bol loop provided it satisfies the right Bol relationAn algebraic loop which is both a left and rightBol loop is called a Moufang loop.Some sources use the term Bol loop to refer to a right Bol loop, whereas some reserve the term for algebraic loops that are Moufang.Although (left and right) Bol loops have relatively weak structural properties, one can show that such structures are power associative. Thus, given an algebraic loop , the element is well-defined for all elements and all integers independent of which order the multiplications are performed...

### Weyl group

Let be a finite-dimensional split semisimple Lie algebra over a field of field characteristic 0, a splitting Cartan subalgebra, and a weight of in a representation of . Thenis also a weight. Furthermore, the reflections with a root, generate a group of linear transformations in called the Weyl group of relative to , where is the algebraic conjugate space of and is the Q-space spanned by the roots (Jacobson 1979, pp. 112, 117, and 119).The Weyl group acts on the roots of a semisimple Lie algebra, and it is a finite group. The animations above illustrate this action for Weyl Group acting on the roots of a homotopy from one Weyl matrix to the next one (i.e., it slides the arrows from to ) in the first two figures, while the third figure shows the Weyl Group acting on the roots of the Cartan matrix of the infinite family of semisimple lie algebras (cf. Dynkin diagram), which is the special linear Lie algebra, ...

### Dimensionality theorem

For a finite group of elements with an th dimensional th irreducible representation,

### Wallpaper groups

The wallpaper groups are the 17 possible plane symmetry groups. They are commonly represented using Hermann-Mauguin-like symbols or in orbifold notation (Zwillinger 1995, p. 260).orbifold notationHermann-Mauguin symbolop12222p2**pmxxpg*2222pmm22*pmg22xpggx*cm2*22cmm442p4*442p4m4*2p4g333p3*333p3ml3*3p3lm632p6*632p6mPatterns created with Artlandia SymmetryWorks for each of these groups are illustrated above.Beautiful patterns can be created by repeating geometric and artistic motifs according to the symmetry of the wallpaper groups, as exemplified in works by M. C. Escher and in the patterns created by I. Bakshee in the Wolfram Language using Artlandia, illustrated above.For a description of the symmetry elements present in each space group, see Coxeter (1969, p. 413)...

### Octahedral group

is the point group of symmetries of the octahedron having order 48 that includes inversion. It is also the symmetry group of the cube, cuboctahedron, and truncated octahedron. It has conjugacy classes 1, , , , , , , , , and (Cotton 1990). Its multiplication table is illustrated above. The octahedral group is implemented in the Wolfram Language as FiniteGroupData["Octahedral", "PermutationGroupRepresentation"] and as a point group as FiniteGroupData["CrystallographicPointGroup", "Oh", "PermutationGroupRepresentation"].The great rhombicuboctahedron can be generated using the matrix representation of using the basis vector .The octahedral group has a pure rotation subgroup denoted that is isomorphic to the tetrahedral group . is of order 24 and has conjugacy classes 1, , , , and (Cotton 1990, pp. 50 and 434). Its multiplication table is illustrated above. The pure..

### Vierergruppe

The vierergruppe is the Abelian abstract group on four elements that is isomorphic to the finite group C2×C2 and the dihedral group . The multiplication table of one possible representation is illustrated below. It can be generated by the permutations 1, 2, 3, 4, 2, 1, 4, 3, 3, 4, 1, 2, and 4, 3, 2, 1.It has subgroups , , , , and all of which are normal, so it is not a simple group. Each element is in its own conjugacy class.

### O'nan group

The O'Nan group is the sporadic group O'Nof order(1)(2)It is implemented in the Wolfram Languageas ONanGroupON[].

### Dihedral group d_6

The dihedral group gives the group of symmetries of a regular hexagon. The group generators are given by a counterclockwise rotation through radians and reflection in a line joining the midpoints of two opposite edges. If denotes rotation and reflection, we have(1)From this, the group elements can be listed as(2)The conjugacy classes of are given by(3)The set of elements which by themselves make up conjugacy classes are in the center of , denoted , so(4)The commutator subgroup is given by(5)which can be used to find the Abelianization. The set of all left cosets of is given by(6)(7)Thus we appear to have two generators for this group, namely and . Therefore, Abelianization gives .It is also known that where is the symmetric group. Furthermore where is the dihedral group with 6 elements, i.e., the group of symmetries of an equilateral triangle.There are thus two ways to produce the character table, either inducing from and using the orthogonality..

### Trivial group

The trivial group, denoted or , sometimes also called the identity group, is the unique (up to isomorphism) group containing exactly one element , the identity element. Examples include the zero group (which is the singleton set with respect to the trivial group structure defined by the addition ), the multiplicative group (where ), the point group , and the integers modulo 1 under addition. When viewed as a permutation group on letters, the trivial group consists of the single element which fixes each letter.The trivial group is (trivially) Abelian and cyclic.The multiplication table for is given below. 111The trivial group has the single conjugacy class and the single subgroup .

### Dihedral group d_5

The group is one of the two groups of order 10. Unlike the cyclic group , is non-Abelian. The molecule ruthenocene belongs to the group , where the letter indicates invariance under a reflection of the fivefold axis (Arfken 1985, p. 248). has cycle index given byIts multiplication table is illustrated above.The dihedral group has conjugacy classes , , , and . It has 8 subgroups: , , , , , , , and , of which , , and , , , , , , , , , are normal.

### Triangular symmetry group

Given a triangle with angles (, , ), the resulting symmetry group is called a triangle group (also known as a spherical tessellation). In three dimensions, such groups must satisfyand so the only solutions are , , , and (Ball and Coxeter 1987). The group gives rise to the semiregular planar tessellations of types 1, 2, 5, and 7. The group gives hyperbolic tessellations.

### Monstrous moonshine

In 1979, Conway and Norton discovered an unexpected intimate connection between the monster group and the j-function. The Fourier expansion of is given by(1)(OEIS A000521), where and is the half-period ratio, and the dimensions of the first few irreducible representations of are 1, 196883, 21296876, 842609326, ... (OEIS A001379).In November 1978, J. McKay noticed that the -coefficient 196884 is exactly one more than the smallest dimension of nontrivial representations of the (Conway and Norton 1979). In fact, it turns out that the Fourier coefficients of can be expressed as linear combinations of these dimensions with small coefficients as follows:(2)(3)(4)(5)Borcherds (1992) later proved this relationship, which became known as monstrous moonshine. Amazingly, there turn out to be yet more deep connections between the monster group and the j-function...

### Dihedral group d_4

The dihedral group is one of the two non-Abelian groups of the five groups total of group order 8. It is sometimes called the octic group. An example of is the symmetry group of the square.The cycle graph of is shown above. has cycle index given by(1)Its multiplication table is illustrated above. has representation(2)(3)(4)(5)(6)(7)(8)(9)Conjugacy classes include , , , , and . There are 10 subgroups of : , , , , , , , , and , . Of these, , , , , , and are normal

### Monster group

The monster group is the highest order sporadic group . It has group order(1)(2)where the divisors are precisely the 15 supersingularprimes (Ogg 1980).The monster group is also called the friendly giant group. It was constructed in 1982 by Robert Griess as a group of rotations in -dimensional space.It is implemented in the Wolfram Languageas MonsterGroupM[].

### Dihedral group d_3

The dihedral group is a particular instance of one of the two distinct abstract groups of group order 6. Unlike the cyclic group (which is Abelian), is non-Abelian. In fact, is the non-Abelian group having smallest group order.Examples of include the point groups known as , , , , the symmetry group of the equilateral triangle (Arfken 1985, p. 246), and the permutation group of three objects (Arfken 1985, p. 249).The cycle graph of is shown above. has cycle index given by(1)Its multiplication table is illustrated above and enumerated below, where 1 denotes the identity element. Equivalent but slightly different forms are given by (Arfken 1985, p. 247) and Cotton (1990, p. 12), the latter of which denotes the abstract group of by .11111111Like all dihedral groups, a reducible two-dimensional representation using real matrices has generators given by and , where is a rotation by radians about an axis passing through the..

### Modulo multiplication group

A modulo multiplication group is a finite group of residue classes prime to under multiplication mod . is Abelian of group order , where is the totient function.A modulo multiplication group can be visualized by constructing its cycle graph. Cycle graphs are illustrated above for some low-order modulo multiplication groups. Such graphs are constructed by drawing labeled nodes, one for each element of the residue class, and connecting cycles obtained by iterating . Each edge of such a graph is bidirected, but they are commonly drawn using undirected edges with double edges used to indicate cycles of length two (Shanks 1993, pp. 85 and 87-92).The following table gives the modulo multiplication groups of small orders, together with their isomorphisms with respect to cyclic groups .groupelements2121, 221, 341, 2, 3, 421, 561, 2, 3, 4, 5, 641, 3, 5, 761, 2, 4, 5, 7, 841, 3, 7, 9101, 2, 3, 4, 5, 6, 7, 8, 9, 1041, 5, 7, 11121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,..

### Thompson group

The Thompson group is the sporadic group Thof order(1)(2)It is implemented in the Wolfram Languageas ThompsonGroupTh[].

### Mclaughlin group

The McLaughlin group is the sporadic group McLof order(1)(2)It is implemented in the Wolfram Languageas McLaughlinGroupMcL[].

### Dihedral group

The dihedral group is the symmetry group of an -sided regular polygon for . The group order of is . Dihedral groups are non-Abelian permutation groups for .The th dihedral group is represented in the Wolfram Language as DihedralGroup[n].One group presentation for the dihedral group is .A reducible two-dimensional representation of using real matrices has generators given by and , where is a rotation by radians about an axis passing through the center of a regular -gon and one of its vertices and is a rotation by about the center of the -gon (Arfken 1985, p. 250).Dihedral groups all have the same multiplication table structure. The table for is illustrated above.The cycle index (in variables , ..., ) for the dihedral group is given by(1)where(2)is the cycle index for the cyclic group , means divides , and is the totient function (Harary 1994, p. 184). The cycle indices for the first few are(3)(4)(5)(6)(7)Renteln and Dundes (2005) give..

### Tetrahedral group

The tetrahedral group is the point group of symmetries of the tetrahedron including the inversion operation. It is one of the 12 non-Abelian groups of order 24. The tetrahedral group has conjugacy classes 1, , , , and (Cotton 1990, pp. 47 and 434). Its multiplication table is illustrated above. The tetrahedral group is implemented in the Wolfram Language as FiniteGroupData["Tetrahedral", "PermutationGroupRepresentation"] and as a point group as FiniteGroupData["CrystallographicPointGroup", "Td", "PermutationGroupRepresentation"]. has a pure rotational subgroup of order 12 denoted (Cotton 1990, pp. 50 and 433). It is isomorphic to the alternating group and has conjugacy classes 1, , , and . It has 10 subgroups: one of length 1, three of length 2, 4 of length 3, one of length 4, and one of length 12. Of these, only the trivial subgroup, subgroup of order 4, and complete..

### Mathieu groups

The five Mathieu groups , , , , and were the first sporadic groups discovered, having been found in 1861 and 1873 by Mathieu. Frobenius showed that all the Mathieu groups are subgroups of .The sporadic Mathieu groups are implemented in the Wolfram Language as MathieuGroupM11[], MathieuGroupM12[], MathieuGroupM22[], MathieuGroupM23[], and MathieuGroupM24[].All the sporadic Mathieu groups are multiply transitive. The following table summarizes some properties of the Mathieu groups, where indicates the transitivity and is the length of the minimal permutation support (from which the groups derive their designations).grouporderfactorization41179205129504032244352042310200960524244823040The Mathieu groups are most simply defined as automorphism groups of Steiner systems, as summarized in the following table.Mathieu groupSteiner system..

### Cyclic group c_12

The cyclic group is one of the two Abelian groups of the five groups total of group order 12 (the other order-12 Abelian group being finite group C2×C6). Examples include the modulo multiplication groups of orders and 26 (which are the only modulo multiplication groups isomorphic to ).The cycle graph of is shown above. The cycle index isIts multiplication table is illustrated above.The numbers of elements satisfying for , 2, ..., 12 are 1, 2, 3, 4, 1, 6, 1, 4, 3, 2, 1, 12.Because the group is Abelian, each element is in its own conjugacy class. There are six subgroups: , , , and . , and which, because the group is Abelian, are all normal. Since has normal subgroups other than the trivial subgroup and the entire group, it is not a simple group.

### Symmetric group

The symmetric group of degree is the group of all permutations on symbols. is therefore a permutation group of order and contains as subgroups every group of order .The th symmetric group is represented in the Wolfram Language as SymmetricGroup[n]. Its cycle index can be generated in the Wolfram Language using CycleIndexPolynomial[SymmetricGroup[n], x1, ..., xn]. The number of conjugacy classes of is given , where is the partition function P of . The symmetric group is a transitive group (Holton and Sheehan 1993, p. 27).For any finite group , Cayley's group theorem proves is isomorphic to a subgroup of a symmetric group.The multiplication table for is illustrated above.Let be the usual permutation cycle notation for a given permutation. Then the following table gives the multiplication table for , which has elements.(1)(2)(3)(1)(23)(3)(12)(123)(132)(2)(13)(1)(2)(3)(1)(2)(3)(1)(23)(3)(12)(123)(132)(2)(13)(1)(23)(1)(23)(1)(2)(3)(132)(2)(13)(3)(12)(123)(3)(12)(3)(12)(123)(1)(2)(3)(1)(23)(2)(13)(132)(123)(123)(3)(12)(2)(13)(132)(1)(2)(3)(1)(23)(132)(132)(2)(13)(1)(23)(1)(2)(3)(123)(3)(12)(2)(13)(2)(13)(132)(123)(3)(12)(1)(23)(1)(2)(3)This..

### Lyons group

The Lyons group is the sporadic group Lyof order(1)(2)It is implemented in the Wolfram Languageas LyonsGroupLy[].

### Cyclic group c_11

The cyclic group is unique group of group order 11. An example is the integers modulo 11 under addition (). No modulo multiplication group is isomorphic to . Like all cyclic groups, is Abelian.The cycle graph of is shown above. The cycle index isIts multiplication table is illustrated above.Because the group is Abelian, each element is in its own conjugacy class. Because it is of prime order, the only subgroups are the trivial group and entire group. is therefore a simple group, as are all cyclic graphs of prime order.

### Sylow theorems

Let be a prime number, a finite group, and the order of . 1. If divides , then has a Sylow p-subgroup. 2. In a finite group, all the Sylow p-subgroups are conjugate for some fixed . 3. The number of Sylow p-subgroups for a fixed is congruent to 1 (mod ).

### Cyclic group c_10

The cyclic group is the unique Abelian group of group order 10 (the other order-10 group being the non-Abelian ). Examples include the integers modulo 10 under addition () and the modulo multiplication groups and (with no others). Like all cyclic groups, is Abelian.The cycle graph of is shown above. The cycle index isIts multiplication table is illustrated above.The numbers of elements satisfying for , 2, ..., 10 are 1, 2, 1, 2, 5, 2, 1, 2, 1, 10.Because the group is Abelian, each element is in its own conjugacy class. There are four subgroups: , , , and . Because the group is Abelian, these are all normal. Since has normal subgroups other than the trivial subgroup and the entire group, it is not a simple group.

### Cyclic group c_9

The cyclic group is one of the two Abelian groups of group order 9 (the other order-9 Abelian group being ; there are no non-Abelian groups of order 9). An example is the integers modulo 9 under addition (). No modulo multiplication group is isomorphic to . Like all cyclic groups, is Abelian.The cycle graph of is shown above. The cycle index isIts multiplication table is illustrated above.The numbers of elements satisfying for , 2, ..., 9 are 1, 1, 3, 1, 1, 3, 1, 1, 9.Because the group is Abelian, each element is in its own conjugacy class. There are three subgroups: , and . Because the group is Abelian, these are all normal. Since has normal subgroups other than the trivial subgroup and the entire group, it is not a simple group.

### Suzuki group

The Suzuki group is the sporadic group Suzof order(1)(2)It is implemented in the Wolfram Languageas SuzukiGroupSuz[].

### Cyclic group c_8

The cyclic group is one of the three Abelian groups of the five groups total of group order 8. Examples include the integers modulo 8 under addition () and the residue classes modulo 17 which have quadratic residues, i.e., under multiplication modulo 17. No modulo multiplication group is isomorphic to .The cycle graph of is shown above. The cycle index isIts multiplication table is illustrated above.The elements satisfy , four of them satisfy , and two satisfy .Because the group is Abelian, each element is in its own conjugacy class. There are four subgroups: , , , and which, because the group is Abelian, are all normal. Since has normal subgroups other than the trivial subgroup and the entire group, it is not a simple group.

### Kronecker decomposition theorem

Every finite Abelian group can be written as a group direct product of cyclic groups of prime power group orders. In fact, the number of nonisomorphic Abelian finite groups of any given group order is given by writing aswhere the are distinct prime factors, thenwhere is the partition function. This gives 1, 1, 1, 2, 1, 1, 1, 3, 2, ... (OEIS A000688).More generally, every finitely generated Abelian group is isomorphic to the group direct sum of a finite number of groups, each of which is either cyclic of prime power order or isomorphic to . This extension of Kronecker decomposition theorem is often referred to as the Kronecker basis theorem.

### Cyclic group c_7

is the cyclic group that is the unique group of group order 7. Examples include the point group and the integers modulo 7 under addition (). No modulo multiplication group is isomorphic to . Like all cyclic groups, is Abelian.The cycle graph is shown above, and the group hascycle index isThe elements of the group satisfy , where 1 is the identity element.Its multiplication table is illustrated aboveand enumerated below. 111111111Because it is Abelian, the group conjugacy classes are , , , , , , and . Because 7 is prime, the only subgroups are the trivial group and the entire group. is therefore a simple group, as are all cyclic graphs of prime order.

The sporadic groups are the 26 finite simple groups that do not fit into any of the four infinite families of finite simple groups (i.e., the cyclic groups of prime order, alternating groups of degree at least five, Lie-type Chevalley groups, and Lie-type groups). The smallest sporadic group is the Mathieu group , which has order 7920, and the largest is the monster group, which has order .The orders of the sporadic groups given in increasing order are 7920, 95040, 175560, 443520, 604800, 10200960, 44352000, 50232960, ... (OEIS A001228). A summary of sporadic groups, as given by Conway et al. (1985), is given below.nameorderfactorizationMathieu group 7920Mathieu group 95040Janko group 175560Mathieu group 443520Janko group 604800Mathieu group 10200960Higman-Sims group HS44352000Janko group 50232960Mathieu group 244823040McLaughlin group McL898128000Held group He4030387200Rudvalis Group Ru145926144000Suzuki group Suz448345497600O'Nan..

### Kronecker basis theorem

A generalization of the Kronecker decomposition theorem which states that every finitely generated Abelian group is isomorphic to the group direct sum of a finite number of groups, each of which is either cyclic of prime power order or isomorphic to . This decomposition is unique, and the number of direct summands is equal to the group rank of the Abelian group.

### Cyclic group c_6

is one of the two groups of group order 6 which, unlike , is Abelian. It is also a cyclic. It is isomorphic to . Examples include the point groups and , the integers modulo 6 under addition (), and the modulo multiplication groups , , and (with no others).The cycle graph is shown above and has cycleindexThe elements of the group satisfy , where 1 is the identity element, three elements satisfy , and two elements satisfy .Its multiplication table is illustrated aboveand enumerated below. 11111111Since is Abelian, the conjugacy classes are , , , , , and . There are four subgroups of : , , , and which, because the group is Abelian, are all normal. Since has normal subgroups other than the trivial subgroup and the entire group, it is not a simple group.

### Special linear group

Given a ring with identity, the special linear group is the group of matrices with elements in and determinant 1.The special linear group , where is a prime power, the set of matrices with determinant and entries in the finite field . is the corresponding set of complex matrices having determinant . is a subgroup of the general linear group and is a Lie-type group. Both and are genuine Lie groups.

### Cyclic group c_5

is the unique group of group order 5, which is Abelian. Examples include the point group and the integers mod 5 under addition (). No modulo multiplication group is isomorphic to .The cycle graph is shown above, and the cycleindexThe elements satisfy , where 1 is the identity element.Its multiplication table is illustrated aboveand enumerated below. 1111111Since is Abelian, the conjugacy classes are , , , , and . Since 5 is prime, there are no subgroups except the trivial group and the entire group. is therefore a simple group, as are all cyclic graphs of prime order.

### Cyclic group c_4

is one of the two groups of group order 4. Like , it is Abelian, but unlike , it is a cyclic. Examples include the point groups (note that the same notation is used for the abstract cyclic group and the point group isomorphic to it) and , the integers modulo 4 under addition (), and the modulo multiplication groups and (which are the only two modulo multiplication groups isomorphic to it).The cycle graph of is shown above, and the cycle index is given by(1)The multiplication table for this group may be written in three equivalent ways by permuting the symbols used for the group elements (Cotton 1990, p. 11). One such table is illustrated above and enumerated below. 111111The conjugacy classes of are , , , and . In addition to the trivial group and the entire group, also has as a subgroup which, because the group is Abelian, is normal. is therefore not a simple group.Elements of the group satisfy , where 1 is the identity element, and two of the elements satisfy..

### Simple group

A simple group is a group whose only normal subgroups are the trivial subgroup of order one and the improper subgroup consisting of the entire original group. Simple groups include the infinite families of alternating groups of degree , cyclic groups of prime order, Lie-type groups, and the 26 sporadic groups.Since all subgroups of an Abelian group are normal and all cyclic groups are Abelian, the only simple cyclic groups are those which have no subgroups other than the trivial subgroup and the improper subgroup consisting of the entire original group. And since cyclic groups of composite order can be written as a group direct product of factor groups, this means that only prime cyclic groups lack nontrivial subgroups. Therefore, the only simple cyclic groups are the prime cyclic groups. Furthermore, these are the only Abelian simple groups.In fact, the classification theorem of finite groups states that such groups can be classified completely..

### Janko groups

The Janko groups are the four sporadic groups , , and . The Janko group is also known as the Hall-Janko group.The Janko groups are implemented in the Wolfram Language as JankoGroupJ1[], JankoGroupJ2[], JankoGroupJ3[], and JankoGroupJ4[].The following table summarized the group orders ofthe Janko groups.grouporderfactorization1755606048005023296086775571046077562880

### Cyclic group c_3

is the unique group of group order 3. It is both Abelian and cyclic. Examples include the point groups , , and and the integers under addition modulo 3 (). No modulo multiplication groups are isomorphic to .The cycle graph of is shown above, and the cycle index isThe elements of the group satisfy where 1 is the identity element.Its multiplication table is illustrated aboveand enumerated below (Cotton 1990, p. 10). 11111Since is Abelian, the conjugacy classes are , , and . The only subgroups of are the trivial group and the entire group, which are both trivially normal. is therefore a simple group, as are all cyclic graphs of prime order.The irreducible representation (character table)is therefore 11111111

### Icosahedral group

The icosahedral group is the group of symmetries of the icosahedron and dodecahedron having order 120, equivalent to the group direct product of the alternating group and cyclic group . The icosahedral group consists of the conjugacy classes 1, , , , , , , , , and (Cotton 1990, pp. 49 and 436). Its multiplication table is illustrated above. The icosahedral group is a subgroup of the special orthogonal group . The icosahedal group is implemented in the Wolfram Language as FiniteGroupData["Icosahedral", "PermutationGroupRepresentation"].Icosahedral symmetry is possible as a rotational group but is not compatible with translational symmetry. As a result, there are no crystals with this symmetry and so, unlike the octahedral group and tetrahedral group , is not one of the 32 point groups.The great rhombicosidodecahedron can be generated using the matrix representation of using the basis vector , where is the golden..

### Cyclic group c_2

The group is the unique group of group order 2. is both Abelian and cyclic. Examples include the point groups , , and , the integers modulo 2 under addition (), and the modulo multiplication groups , , and (which are the only modulo multiplication groups isomorphic to ).The group is also trivially simple, and forms the subject for the humorous a capella song "Finite Simple Group (of Order 2)" by the Northwestern University mathematics department a capella group "The Klein Four."The cycle graph is shown above, and the cycleindex isThe elements satisfy , where 1 is the identity element.Its multiplication table is illustrated aboveand enumerated below. 1111The conjugacy classes are and . The only subgroups of are the trivial group and entire group , both of which are trivially normal.The irreducible representation for the group is ...

### Rudvalis group

The Rudvalis group is the sporadic group Ruof order(1)(2)It is implemented in the Wolfram Languageas RudvalisGroupRu[].

### Cyclic group

A cyclic group is a group that can be generated by a single element (the group generator). Cyclic groups are Abelian.A cyclic group of finite group order is denoted , , , or ; Shanks 1993, p. 75), and its generator satisfies(1)where is the identity element.The ring of integers form an infinite cyclic group under addition, and the integers 0, 1, 2, ..., () form a cyclic group of order under addition (mod ). In both cases, 0 is the identity element.There exists a unique cyclic group of every order , so cyclic groups of the same order are always isomorphic (Scott 1987, p. 34; Shanks 1993, p. 74). Furthermore, subgroups of cyclic groups are cyclic, and all groups of prime group order are cyclic. In fact, the only simple Abelian groups are the cyclic groups of order or a prime (Scott 1987, p. 35).The th cyclic group is represented in the Wolfram Language as CyclicGroup[n].Examples of cyclic groups include , , , ..., and the modulo multiplication..

### Quaternion group

The quaternion group is one of the two non-Abelian groups of the five total finite groups of order 8. It is formed by the quaternions , , , and , denoted or .1111111111The multiplication table for is illustrated above, where rows and columns are given in the order , , , , 1, , , , as in the table above.The cycle graph of the quaternion group is illustratedabove.The quaternion group has conjugacy classes , , , , and . Its subgroups are , , , , , and , all of which are normal subgroups.

### Held group

The Held group is the sporadic group Heof order(1)(2)It is implemented in the Wolfram Languageas HeldGroupHe[].

### Cycle index

Let denote the number of cycles of length for a permutation expressed as a product of disjoint cycles. The cycle index of a permutation group of order and degree is then the polynomial in variables , , ..., given by the formula(1)The cycle index of a permutation group is implemented as CycleIndexPolynomial[perm, x1, ..., xn], which returns a polynomial in . For any permutation , the numbers satisfy(2)and thus constitutes a partition of the integer . Sets of values are commonly denoted , where ranges over all the -vectors satisfying equation (2).Formulas for the most important permutation groups (the symmetric group , alternating group , cyclic group , dihedral group , and trivial group ) are given by(3)(4)(5)(6)(7)where means divides and is the totient function (Harary 1994, p. 184)...

### Projective symplectic group

The projective symplectic group is the group obtained from the symplectic group on factoring by the scalar matrices contained in that group. is simple except for(1)(2)(3)so it is given the simpler name , with .

### Projective special unitary group

The projective special unitary group is the group obtained from the special unitary group on factoring by the scalar matrices contained in that group. is simple except for(1)(2)(3)so it is given the simpler name , with .