$p$-Groups for which each outer $p$-automorphism centralizes only $p$ elements

TL;DR

This paper characterizes finite p-groups where every outer p-automorphism fixes exactly p elements, showing such groups are of order at most p^2.

Contribution

It provides a complete classification of finite p-groups with the property that each outer p-automorphism centralizes only p elements.

Findings

Groups of order at most p^2 have this automorphism property.

Outer p-automorphisms in larger groups fix more than p elements.

The characterization is complete for all finite p-groups.

Abstract

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 $p^2$.