Let $H$ be the group of integers mod $p$, under addition, where $p$ is prime. Suppose that $n$ is an integer $1 \leq n \leq p$, and let $G$ be the group $H \times H \times \dots \times H$ ($n$ times). Show that $G$ has no automorphism of order $p^2$.
I'm pretty stumped on this one. I don't have a great handle on the order of an automorphism. Relevant facts of this problem to me so far as that this group has $p^n$ elements and every non-identity element has order $p$. Any clarity/intuition around this and hints would be greatly appreciated.
Edit: Thought about it some more and I know that $Aut(Z_p^n) \cong (\mathbb{Z}/p^n\mathbb{Z})^\times$ which is a cyclic group of order $p^{n-1}(p-1)$. If I could show that $p^2$ does not divide $p^{n-1}$ I would be done...