I'm stuck on this exercise from Herstein's topics in algebra:
Suppose that $G$ is a group and $|G| = pm$, where $p \not| ~ m$ and $p$ is a prime. If $H$ is a normal subgroup of order $p$ in $G$, prove that $H$ is characteristic.
How to prove that? What we know:
- $H$ and $\varphi(H)$ have prime order $p$, so they must be cyclic and abelian
- They are both normal subgroups of $G$
how to show that, in fact, we have $H = \varphi(H)$ for every automorphism $\varphi$ ?