We now give one of the most important results in group theory. It shows that every group is isomorphic to a group of permutations. That is, every group is isomorphic to a subgroup of the symmetric group on some suitable set .
Every group is isomorphic to a group of permutations.
The proof of this result is constructive in that it gives you an algorithm for converting a group into a permutation group. For each let be the permutation defined by In other words, is derived from right multiplication of the elements of G by g . That is a permutation is easily established. Let and define by . Now for any we have Hence, and , proving f is a homomorphism.