[math-fun] Group theory question

Let G be a group, and let S be the permutation group on G.  For a in G, let L(a) be the permutation that sends x to ax, and let R(b) be the permutation that sends x to xb' (' denoting inverse).  Then im L and im R are isomorphic copies of G in S, and each is the centralizer in S of the other.  I have a vague memory of having seen a proof of the statement concerning centralizers, but can't pin it down.  Can someone point me to either a proof, or a counterexample?   --  Gene