A note on automorphisms of finite $p$-groups

TL;DR

This paper investigates conditions under which the order of a finite non-cyclic p-group divides the order of its automorphism group, focusing on groups with specific abelian or elementary abelian substructures.

Contribution

It establishes new divisibility conditions for automorphism groups of certain finite p-groups based on their subgroup structures.

Findings

If G has an abelian maximal subgroup, then |G| divides |Aut(G)|.

If G has an elementary abelian center with C_G(Z(Φ(G))) ≠ Φ(G), then |G| divides |Aut(G)|.

Abstract

Let $G$ be a finite non-cyclic $p$-group of order at least $p^3$. If $G$ has an abelian maximal subgroup, or if $G$ has an elementary abelian centre with $C_G(Z(\Phi(G))) \ne \Phi(G)$, then $|G|$ divides $|\text{Aut}(G)|$.