Let be a group. Define to be the symmetry group on , that is, consists of all bijections and under the binary operation of function composition. As you learned is the symmetric group (by definition). We are going to define a function . So for every we need to define so that becomes a permutation on . That is done by defining to be the function that is defined by . Notice that is a permutation of . Therefore, we will define . Notice, that is an element of but is a function on ! Be careful, this sometimes confuses people. The next thing to notice is that if then it must mean that so is one-to-one. Finally, is a homomorphism because . From this point you use the result from group theory that under an injective homomorphism the group is isomorphic to its homomorphic image, so, , but is a subgroup of . Since is a subgroup of a symmetric group it itself is a permutation group. Thus, all groups are isomorphic to permutation groups.