A module-theoretic approach to abelian automorphism groups

TL;DR

This paper introduces a module-theoretic method to construct non-special finite p-groups with abelian automorphism groups, simplifying previous approaches and expanding known examples.

Contribution

It presents a novel module-theoretic framework for generating non-special p-groups with abelian automorphism groups, avoiding complex calculations.

Findings

Constructed new non-special p-groups with abelian automorphism groups

Simplified the process of finding such groups using module theory

Extended the class of known examples beyond special p-groups

Abstract

There are several examples in the literature of finite non-abelian $p$-groups whose automorphism group is abelian. For some time only examples that were special $p$-groups were known, until Jain and Yadav [JY12] and Jain, Rai and Yadav [JRY13] constructed several non-special examples. In this paper we show how a simple module-theoretic approach allows the construction of non-special examples, starting from special ones constructed by several authors, while at the same time avoiding further direct calculations.