Groups in which every non-cyclic subgroup contains its centralizer

TL;DR

This paper investigates groups where every non-cyclic subgroup contains its centralizer, providing classifications for finite p-groups, simple groups, and describing the structure of nilpotent and supersolvable groups with this property.

Contribution

It offers a comprehensive classification of finite p-groups and simple groups with the property, and describes the structure of nilpotent and supersolvable groups in this class.

Findings

Classified finite p-groups with the property

Classified finite simple groups with the property

Described structure of nilpotent and supersolvable groups with the property

Abstract

We study groups having the property that every non-cyclic subgroup contains its centralizer. The structure of nilpotent and supersolvable groups in this class is described. We also classify finite $p$-groups and finite simple groups with the above defined property.