## Every automorphism of an elementary abelian p-group having p-power order has nontrivial fixed points

Let $p$ be a prime, let $V$ be a nontrivial elementary abelian $p$-group, and let $\varphi$ be an automorphism of $V$ of $p$-power order. Prove that $\varphi$ has nontrivial fixed points.

Note that $\langle \varphi \rangle \leq \mathsf{Aut}(V)$; use the inclusion map to construct $P = V \rtimes \langle \varphi \rangle$. Now $P$ is a $p$-group, and $V \leq P$ is normal, so that by Theorem 1, $V \cap Z(P)$ is nontrivial.

Now let $x \in V \cap Z(P)$ with $x \neq 1$. Then $(x, \varphi) = (x,1)(1,\varphi)$ $= (1,\varphi)(x,1)$ $= (\varphi(x),\varphi)$. Comparing entries we see that $x = \varphi(x)$.