Cayley's theorem
In group theory, Cayley's theorem asserts that any group "G" is isomorphic to a subgroup of the symmetric group of "G" (regarded as a set), so that "G" is a permutation group. This theorem generalizes straightforwardly to monoids: any monoid "M" is a transformation monoid of its underlying set, via the action given by left (or right) multiplication. This action is faithful because if "ax" = "bx" for all "x" in "M", then by taking "x" equal to the identity element, we have "a" = "b".