In group theory, Cayley's theorem, named in honor of Arthur Cayley, states that every group G is isomorphic to a subgroup of the symmetric group on G. This can be understood as an example of the group action of G on the elements of G. Group theory is that branch of mathematics concerned with the study of groups. ... Arthur Cayley (August 16, 1821 - January 26, 1895) was a British mathematician. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. ... In mathematics, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group operation... In mathematics, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i. ... In mathematics, groups are often used to describe symmetries of objects. ...

A permutation of a set G is any bijective function taking G onto G; and the set of all such functions forms a group under function composition, called the symmetric group on G, and written as Sym(G). In mathematics, especially in abstract algebra and related areas, a permutation is a bijection, from a finite set X onto itself. ... In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ...

Cayley's theorem puts all groups on the same footing, by considering any group (including infinite groups such as (R,+)) as a permutation group of some underlying set. Thus, theorems which are true for permutation groups are true for groups in general.

Proof of the theorem

From elementary group theory, we can see that for any element g in G, we must have g*G = G; and by cancellation rules, that g*x = g*y if and only if x = y. So multiplication by g acts as a bijective function f_g : G → G, by defining f_g(x) = g*x. Thus, f_g is a permutation of G, and so is a member of Sym(G). First theorems about groups A group (G,*) is usually defined as: G is a set and * is an associative binary operation on G, obeying the following rules (or axioms): A1. ... In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ...

The subset K of Sym(G) defined as K = {f_g : g in G and f_g(x) = g*x for all x in G} is a subgroup of Sym(G) which is isomorphic to G. The fastest way to establish this is to consider the function T : G → Sym(G) with T(g) = f_g for every g in G. T is a group homomorphism because (using "•" for composition in Sym(G)):(f_g • f_h)(x) = f_g(f_h(x)) = f_g(h*x) = g*(h*x) = (g*h)*x = f_(g*h)(x), for all x in G, and hence: T(g) • T(h) = f_g • f_h = f_(g*h) = T(g*h). The homomorphism T is also injective since T(g) = id_G (the identity element of Sym(G)) implies that g*x = x for all x in G, and taking x to be the identity element e of G yields g = g*e = e. Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the identity element... In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ...

Thus G is isomorphic to the image of T, which is the subgroup K considered earlier.

T is sometimes called the regular representation of G.