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