On automorphisms of finite $p$-groups

TL;DR

This paper provides a concise, elementary proof of a known result that for certain finite p-groups, the order of the group divides the order of its automorphism group, specifically when the pair (G, Z(G)) is a Camina pair.

Contribution

It offers a simplified and elementary proof of a previously established theorem relating the size of finite p-groups and their automorphism groups in the Camina pair context.

Findings

The order of G divides the order of Aut(G) for Camina pairs.

The proof is shorter and more elementary than previous proofs.

The result applies to finite p-groups with specific structural properties.

Abstract

It is proved in [J. Group Theory, {\bf 10} (2007), 859-866] that if $G$ is a finite $p$-group such that $(G,Z(G))$ is a Camina pair, then $|G|$ divides $|\Aut(G)|$. We give a very short and elementary proof of this result.