Consider the abelian group $$G = \underbrace{\mathbb{Z}/p\oplus\cdots\oplus \mathbb{Z}/p}_{n},$$ where $p$ is prime and $1\le n \le p$. I want to show that $G$ has no automorphism of order $p^2$.
I observe that group automorphism of $G$ is the same as linear isomorphism of $G$ as a $\mathbb{Z}/p$-vector space. So the question is the same as to prove that there is no $n\times n$ invertible matrix in $\mathbb{Z}/p$ that has order $p^2$.
My attempt is to calculate the order of $GL(n,\mathbb{Z}/p)$ and show that $p^2$ does not divide that order, but unfortunately this only works for $n\le 2$.