Cayley's Theorem - Proof of The Theorem

Proof of The Theorem

Where g is any element of G, consider the function f_g : G → G, defined by f_g(x) = g*x. By the existence of inverses, this function has a two-sided inverse, . So multiplication by g acts as a bijective function. Thus, f_g is a permutation of G, and so is a member of Sym(G).

The set 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)):

for all x in G, and hence:

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. Alternatively, T is also injective since, if g*x=g'*x implies g=g' (by post-multiplying with the inverse of x, which exists because G is a group).

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

T is sometimes called the regular representation of G.