Cayley's Theorem: Any group is isomorphic to a subgroup of a permutations group.

Arthur Cayley was an Irish mathematician. The name Cayley is the Irish name more commonly spelled Kelly.

Proof:

Let S be the set of elements of a group G and let * be its operation.

Now let F be the set of one-to-one functions from the set S to the set S. Such functions are called permutations of the set. The set F with function composition (·) is a group.

Proof of this proposition:

Function composition is closed and associative. There is an identity element e(x)=x for all x belonging to S. There is an inverse for any function: if f(x)=y then f^-1(y)=x. Thus (F, ·) is a group.

Proof of the thorem

For any element g of S consider the function f_g(x)=g*x for all x in S. This function is an element of F.

Consider f_g*h(x). Since G is a group g*h is an element of S and hence f_g*h is an element of F. Furthermore, since * is associative in G,

(g*h)*x = g*(h*x) = g*(f_h(x)) = f_g(f_h(x)) = f_g·f_h(x)
but
(g*h)*x is f_g*h(x)
so
f_g*h = f_g·f_h

Therefore the set {f_g for all g in G} is a subgroup of F. Thus G is isomorphic to a subgroup of F with the operation function composition, (·).