Cayley's theorem
Any group "G" can act on itself (the elements of the group being thought of as the set "M") in many ways. In particular, there is a regular action given by (left) multiplication in the group. That is, "f"("g", "x") = "gx" for all "g" and "x" in "G". For each fixed "g", the function "f""g"("x") = "gx" is a bijection on "G" and therefore a permutation of the set of elements of "G". Each element of "G" can be thought of as a permutation in this way and so "G" is isomorphic to a permutation group; this is the content of Cayley's theorem.