Groups in which every non-abelian subgroup is self-normalized

TL;DR

This paper classifies finite and infinite soluble groups where every non-abelian subgroup is equal to its normalizer, addressing a problem related to group self-normalization properties.

Contribution

It provides a complete classification of finite groups with this property and describes all infinite soluble groups exhibiting it.

Findings

Finite groups with the property are fully classified.

All infinite soluble groups with the property are characterized.

The work relates to an open problem by Berkovich.

Abstract

We study groups having the property that every non-abelian subgroup is equal to its normalizer. This class of groups is closely related to an open problem posed by Berkovich. We give a full classification of finite groups having the above property. We also describe all infinite soluble groups in this class.