  An automorphism of a group is called outer if it is not an inner
automorphism.
  Let $G$ be a finite $p$-group. Then for every outer $p$-automorphism $\phi$
of $G$ the subgroup $C_G(\phi)=\{x\in G \;|\; x^\phi=x\}$ has order $p$ if and
only if $G$ is of order at most $p^2$.
