Cayley's theorem

In group theory, Cayley's theorem, named in honor of Arthur Cayley, states that every group 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.

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).

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).

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.

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.