A classification of the finite non-solvable minimal non-CA-groups

TL;DR

This paper classifies finite non-solvable minimal non-CA-groups, which are groups where the centralizer of every non-central element is abelian, but the group itself is not, and all proper subgroups are CA-groups.

Contribution

It provides a comprehensive classification of finite non-solvable minimal non-CA-groups, a previously uncharacterized class of groups.

Findings

Complete classification of finite non-solvable minimal non-CA-groups

Identification of structural properties distinguishing these groups

Clarification of the relationship between CA-groups and their minimal non-CA counterparts

Abstract

A group is called a CA-group if the centralizer of every non-central element is abelian. Furthermore, a group is called a minimal non-CA-group if it is not a CA-group itself, but all of its proper subgroups are. In this paper, we give a classification of the finite non-solvable minimal non-CA-groups.