# Cayley's Theorem

__Cayley’s Theorem__**:**

Every group is isomorphic to a permutation group.

__Proof__**: **Let be a finite group of order . If , then , . Now consider a function from into , defined by

For . Therefore, the function is one-one.

The function is also onto because if is any element of then there exists an element such that

Thus is one-one from onto . Therefore, is a permutation on . Let denote the set of all such one-to-one functions defined on corresponding to every element of , i.e.

Now, we show that is a group with respect to the product of functions.

**(i) Closure Axiom:** Let where, then

Since , therefore and thus is closed under the product of functions.

**(ii) Associative Axiom:** Let where , then

The product of functions is associative in .

**(iii) Identity Axiom:** If is the identity element in , then is the identity of because we have and .

**(iv) Inverse Element:** If is the inverse of in , then is the inverse of in because and

Hence is a group with respect to the composite of functions denoted by the symbol .

Now consider the function and into defined by .

is one-one because for .

is onto because if then for , we have

preserves composition in and because if then

Hence .