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).

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