A cohomological property of semi-abelian $p$-groups

TL;DR

This paper proves a cohomological property for semi-abelian p-groups, showing they have non-inner automorphisms of order p and clarifying their power structure, thus resolving longstanding open questions.

Contribution

It introduces a cohomological property for semi-abelian p-groups and demonstrates their automorphism and power structure characteristics, settling previous open problems.

Findings

Semi-abelian p-groups have non-inner automorphisms of order p.

The power structure of semi-abelian p-groups is clarified.

A cohomological property characterizes semi-abelian p-groups.

Abstract

We prove a cohomological property for a class of finite $p$-groups introduced earlier by M. Y. Xu, which we call semi-abelian $p$-groups. This result implies that a semi-abelian $p$-group has non-inner automorphisms of order $p$, which settles a longstanding problem for this class. We answer also, independetly, an old question of M. Y. Xu about the power structure of semi-abelian $p$-groups.