Let be a bijection. Define by for all and prove the following.
- is well defined; that is, if is a permutation of then is a permutation of .
- is a bijection. (Find a two-sided inverse.)
- is a homomorphism; that is, .
- Let . Note that , and that the composition of bijections is a bijection, so that is in fact a permutation of .
- Define by . Then , and similarly . So is a two-sided inverse for , and thus is a bijection.
- We have , hence is a homomorphism.