Cayley's Theorem states that every group is isomorphic to a permutation group, i.e., a subgroup of a symmetric group; in other words, every group acts on some set. Although the result is simple, it is deep, as it characterizes group structure as the structure of a family of bijections.
We prove that each group is isomorphic to a group of bijections on itself. Indeed, for all , let be the mapping from into itself. Then is a bijection, for all ; and for all , . Thus is isomorphic to the set of permutations on .
The action of on itself as described in the proof is called the left action of on itself. Right action is defined similarly.