Equality of certain automorphism groups of finite $p$-groups

TL;DR

This paper investigates conditions under which specific subgroups of automorphism groups of finite p-groups are equal, providing new results and simplified proofs in this area.

Contribution

It introduces a technical lemma that leads to new conditions and streamlined proofs for subgroup equalities within automorphism groups of finite p-groups.

Findings

Established new necessary and sufficient conditions for subgroup equalities

Provided simplified proofs of existing results

Extended understanding of automorphism group structures

Abstract

Let $G$ be a finite $p$-group and let Aut$(G)$ denote the full automorphism group of $G$. In the recent past, there has been interest in finding necessary and sufficient conditions on $G$ such that certain subgroups of Aut$(G)$ are equal. We prove a technical lemma and, as a consequence, obtain some new results and short and alternate proofs of some known results of this type.