On finite $p$-groups whose automorphisms are all central

TL;DR

This paper constructs specific finite p-groups with unique automorphism properties, including examples with abelian automorphism groups that are not special, and groups where all automorphisms are central, addressing longstanding conjectures.

Contribution

It provides counterexamples to Mahalanobis's conjecture and constructs groups with all automorphisms central, solving a problem posed by Malinowska.

Findings

Constructed non-special finite p-groups with abelian automorphism groups.

Provided examples of finite p-groups with all automorphisms central.

Countered existing conjectures about automorphism group structures.

Abstract

An automorphism $\alpha$ of a group $G$ is said to be central if $\alpha$ commutes with every inner automorphism of $G$. We construct a family of non-special finite $p$-groups having abelian automorphism groups. These groups provide counter examples to a conjecture of A. Mahalanobis [Israel J. Math., {\bf 165} (2008), 161 - 187]. We also construct a family of finite $p$-groups having non-abelian automorphism groups and all automorphisms central. This solves a problem of I. Malinowska [Advances in group theory, Aracne Editrice, Rome 2002, 111-127].