
Encyclopaedia of DesignTheory: Topic Essay 
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The symmetry of an object is specified by its structurepreserving mappings and the manner in which they compose with one another. It is this notion of a set with a composition which is the basis of the definition.
A binary operation on a set G (typically denoted by a symbol like o) is a function from G×G to G. We write the value of the function on the pair (g,h ) in G×G (the result of "composing" g and h) as g o h. A group is a set G with a binary operation o satisfying the following conditions:
If it satisfies the additional condition
The set of symmetries of a mathematical object (suitably defined) always has the structure of a group, where the operation is composition. For the composition of two symmetries is a symmetry; the identity map is a symmetry; a symmetry is a onetoone and onto map, and so has an inverse, which is also a symmetry; and composition of maps is always associative.
Symmetry groups can be generalised as follows. A permutation group is a set G of permutations (onetoone and onto maps) of a set Omega which is closed under composition, contains the identity map, and contains the inverse of each of its elements. (A permutation group is a group: the associative law is again automatic.) Thus, each symmetry group is a permutation group. A lot of work has gone into deciding which permutation groups are symmetry groups of objects of particular types such as graphs or designs. (Every permutation group is the symmetry group of some suitably constructed object.)
Cayley's Theorem shows that, conversely, every group can be represented as a permutation group. The proof is as follows. (This argument is stated for finite groups but works more generally.)
Let G={g_{1},g_{2},...,g_{n}} be a group. The Cayley table of G is the n×n matrix with (i,j) entry k if g_{i} o g_{j}=g_{k}. It follows from the group axioms (G2) and (G3) that the Cayley table is a Latin square. Thus, each row is a permutation of {1,...,n}. Now it can be checked that, if pi_{i} is the permutation corresponding to the ith row, then pi_{i} o pij=pi_{k} if and only if g_{i} o g_{j}=g_{k}. (Here the operation on permutations is composition.) Thus the permutations form a group identical to G.
We say that a group G with operation o and a group H with operation * are isomorphic if there is a onetoone correspondence from G to H so that, if g_{1} corresponds to h_{1} and g_{2} to h_{2}, then g_{1} o g_{2} corresponds to h_{1} * h_{2}. Isomorphic groups are "the same" from an algebraic point of view, even though their elements may be quite different. Thus, Cayley's Theorem really states:
Theorem 1 Every group is isomorphic to a permutation group.
The order of a group is the number of elements in the group. It may be finite or infinite, but we will be mainly concerned with finite groups.
Two very different examples of groups, one infinite and abelian, the other finite and (almost always) nonabelian:
[Sometimes instead we think of the operation as "addition", and write g_{1} + g_{2}. This is especially common when the group is abelian. In this case, we write the identity as 0, and the inverse of g as g.]
Let G be a group. A subgroup of G is a nonempty subset which forms a group in its own right, with respect to the operation inherited from G. That is, H must satisfy the conditions
In fact the second condition follows from the others, and all follow from the single condition
We write H<=G to denote that H is a subgroup of G.
Let H be a subgroup of G. The relation ~_{r} on G defined by
x~_{r} y if and only if xy^{1} in His an equivalence relation on H. Its equivalence classes are called right cosets of H in G, and are sets of the form
Hx = {hx : h in H}.The element x is called a right coset representative for the right coset Hx.
Dually, the relation ~_{l} given by
x~_{l} y if and only if x^{1}y in His an equivalence relation, whose equivalence classes are called left cosets of H in G, and have the form
xH = {xh : h in H}.
The left and right cosets of a given subgroup may give different partitions of the group. But the number of elements in a coset of either type is equal to the number of elements in the subgroup. (For right cosets, the correspondence h <> hx is a bijection between H and Hx. So the number of cosets of either type (the index of H in G) is equal to G/H. We deduce Lagrange's Theorem:
Theorem 2 The order of a subgroup H of a finite group G divides the order of G.
The converse of Lagrange's Theorem is false; if G=n and m divides n, there may be no subgroup of order m in G. One case where such a subgroup exists is given by Sylow's Theorem, one of the most important theorems in finite group theory.
Theorem 3 Let G be a group of order n=p^{a}.b, where p is prime and p does not divide b. Then
A subgroup whose order is the exact power of the prime p which divides G is called a Sylow psubgroup of G.
If H is a normal subgroup of G, then we can define an operation on the set G/H of (left or right) cosets of H in G by the rule
Hx o Hy = H(xy).(Of course it is necessary to show that the definition doesn't depend on the choice of coset representatives x and y.) It can be shown that, with this operation, G/H is a group. This group is called the factor group or quotient group of G by H.
How do normal subgroups arise "in nature"?
A homomorphism from a group G to a group H is a function F : G>H with the property that
F(g_{1}g_{2}) = F(g_{1})F(g_{2})for all g_{1},g_{2} in G.
Perhaps the most familiar example of a homomorphism is the function from the additive group Z of integers to the group Z/nZ of integers modulo n, for some positive integer n, which maps each integer k to the congruence class k (mod n.
Another example is the sign map from the symmetric group S_{n} to the multiplicative group {+1,1}, which maps each permutation to its sign. (The sign of a permutation g in S_{n} is (1)^{nc(g)}, where c(g) is the number of cycles of g, including fixed points.)
The kernel of a homomorphism F is the set
Ker(F) = {g in G : F(g)=1_{H}}of elements of G mapped to the identity element of H. The image is, as usual, the set
Im(F) = {F(g) : g in G}of elements of H to which some element of G is mapped. These are described by the Isomorphism Theorem:
Theorem 4 Let F be a homomorphism from G to H. Then
Thus we may say simply
A normal subgroup is the kernel of a homomorphism.
An example of a simple group is the cyclic group C_{p} of prime order p, consisting of elements x^{i} for i=0,...,p1, with composition
x^{i}x^{j} = x^{i+j mod p}.By Lagrange's Theorem, this group has no nontrivial subgroups at all!
If G is composite, with a nontrivial normal subgroup H, then we can often reduce questions about G to questions about the smaller groups H and G/H. If either of these is composite, we can continue the process. Eventually we reach a series
{1} = G_{0} <= G_{1} <= ... <= G_{r} = Gwhich cannot be further refined. Thus, for i=1,...,r, we have that G_{i1} is a normal subgroup of G_{i}, and G_{i}/G_{i1} is simple. Such a series is called a composition series of G, and the simple groups G_{i}/G_{i1} are the composition factors. We are only interested in the composition factors up to isomorphism; they form a multiset, since a given simple group may be isomorphic to G_{i}/G_{i1} for several values of i.
The JordanHölder Theorem states:
Theorem 5 Any two composition series of a finite group G give rise to the same multiset of composition factors.
In a sense, this reduces the study of finite groups to two parts:
To indicate just how far we are from a solution of the second problem, here are some computational results obtained recently by Besche, Eick and O'Brien. The number of groups of order 2000 or less is 49,910,529,484. Of these, more than 99% have order 1024=2^{10}. However, for every group of order 2^{10}, the list of composition factors consists of a single group (the cyclic group of order 2) with multiplicity 10. There is a sense in which the most complicated groups are those of primepower order; such a group has just one composition factor (cyclic of prime order) with the appropriate multiplicity.
However, the first part of the problem has been solved as a result of a major collaborative effort. We proceed to discuss this.
Even the detailed statement of the theorem cannot be given here. Essentially the result is as follows.
Theorem 6 A finite simple group is of one of the following types:
We have already seen the cyclic groups of prime order. Here is a brief description of the remaining groups.
The alternating group A_{n} consists of all even permutations of the set {1,...,n}. We saw earlier that it is the kernel of the sign homomorphism from the symmetric group S_{n} to C_{2}, so it is a normal subgroup of S_{n}. Galois showed that, for n>=5, the alternating group A_{n} is simple (so that the composition factors of S_{n} are A_{n} and C_{2}).
Groups of Lie type are harder to describe. They are closely related to certain matrix groups over finite fields. They fall into a number of families,of which the simplest consists of the projective special linear groups PSL(n,q)=SL(n,q)/Z, where SL(n,q) consists of all matrices of determinant 1, and Z is the normal subgroup consisting of scalar matrices. Further families correspond to other "classical" groups (symplectic, orthogonal and unitary) over finite fields, and there are some "exceptional" families constructed from exceptional Lie algebras or automorphisms of other groups. Carter's book [3] gives details.
The 26 sporadic groups have no uniform definition, but were constructed individually. See the ATLAS [4] for details.
First, a brief reminder about terminology. A permutation group on the set {1,...,n} is a subgroup of the symmetric group S_{n} (that is, a group whose elements are permutations and whose operation is composition). The number n is its degree. A permutation group G is
Among the consequences of CFSG are the following:
A permutation group often arises in practice as the automorphism group of some structure (graph, design, etc.) The program of choice for testing isomorphism of graphs and other combinatorial objects, and for calculating their automorphism groups, is nauty [6]. The GAP share package GRAPE includes an interface with nauty; the automorphism groups of graphs returned by the latter can be handled directly in GAP. The forthcoming package DESIGN will extend this functionality to designs.
In the remainder of this essay we sketch briefly how permutation groups are handled in a computer.
A group is usually input to the computer by giving a set of permutations which generate it. Now given a set of generators, the orbit of a point x (the set of all images of x under elements of G) can be computed by an algorithm similar to that for finding a connected component of a graph: starting with x, add in any point which is the image of an existing point under a generator until the resulting set is closed under all generators. The procedure implicitly finds coset representatives for the stabiliser of x: such a set consists of one element mapping x to each possible image.
Now Schreier's Lemma provides an algorithm which, given generators for a group and coset representatives for a subgroup, finds generators for the subgroup. So we can compute generators for the subgroup G_{1} fixing x.
Continuing this process, we find a sequence
G = G_{0} >= G_{1} >= ... >= G_{d} = {1}of subgroups of G, where G_{i} is the stabiliser of points x_{1},...,x_{i}, for 1<=i<=d. (Unlike in the JordanHölder theorem, these subgroups are not necessarily normal.) At this point we can calculate the order of G and can test any permutation for membership in G. Moreover, an element of G is uniquely determined by the images of x_{1},...,x_{d}, so arbitrary elements of G can be represented in more compact form.
Using this representation, the packages enable the user to compute any grouptheoretical properties of interest, including (but far from limited to) Sylow subgroups, composition factors, images of homomorphisms, etc.
It should be mentioned that groups can be handled in other ways too. Instead of permutation generators, we may give matrix generators, or abstract generators and defining relations.