1. The problem statement, all variables and given/known data This is question 30, section 2.5 from "Abstract Algebra 3rd edition" by Herstein. 2. Relevant information A subgroup H of group G is called characteristic if for all automorphisms phi of G, phi(H) is a subset of H. (I paraphrased this from question 29. I don't know why he says "phi(H) is a subset of H" but not "phi(H) and H are the same set". After all, phi is bijective?) 3. The attempt at a solution Here is what I did: H is a subgroup of order prime, so it is cyclic, generated by any non-identity element in H. Call H <a>. Let phi be an arbitrary automorphism on G, then phi must map <a> to yet another cyclic subgroup of order p in G, call this <phi(a)>. Since both <a> and <phi(a)> are cyclic, they're either the same subgroup, or only share the identity element. Let K of be the smallest subgroup generated by every possible products of <a> and <phi(a)>. If <a> and <phi(a)> are different, then the order of K is p2. So p2 has to divide pm, which contradicts the fact that p does not divide m. So <a> and <phi(a)> have to be the same subgroup in G. Hence any automorphism phi on G will also be an automorphism on <a>, which implies that <a>, aka H, is characteristic. My problem: I never used the fact that H is a normal subgroup. I'm being quite confused. Where was I wrong? Many thanks for the help.