On finite $p$-groups with abelian automorphism group

TL;DR

This paper constructs specific finite p-groups with abelian automorphism groups, exploring their subgroup relations and providing examples that satisfy particular conditions, advancing understanding of automorphism groups in p-groups.

Contribution

It introduces new constructions of finite p-groups with abelian automorphism groups and characterizes their subgroup relations, including examples with specific properties.

Findings

Constructed p-groups with abelian automorphism groups where a3_2(G) < Z(G) < \u03a6(G)

Proved conditions for groups with elementary abelian automorphism groups

Provided examples where exactly one of two key subgroup conditions holds

Abstract

We construct, for the first time, various types of specific non-special finite $p$-groups having abelian automorphism group. More specifically, we construct groups $G$ with abelian automorphism group such that $\gamma_2(G) < \mathrm{Z}(G) < \Phi(G)$, where $\gamma_2(G)$, $\mathrm{Z}(G)$ and $\Phi(G)$ denote the commutator subgroup, the center and the Frattini subgroup of $G$ respectively. For a finite $p$-group $G$ with elementary abelian automorphism group, we show that at least one of the following two conditions holds true: (i) $\mathrm{Z}(G) = \Phi(G)$ is elementary abelian; (ii) $\gamma_2(G) = \Phi(G)$ is elementary abelian, where $p$ is an odd prime. We construct examples to show the existence of groups $G$ with elementary abelian automorphism group for which exactly one of the above two conditions holds true.