Finite groups with small automizers for every abelian subgroup of non prime power order

TL;DR

This paper investigates finite groups where the centralizer equals the normalizer for every abelian subgroup of non-prime power order, and classifies all such nilpotent and minimal non-nilpotent groups.

Contribution

It introduces a classification of finite groups with specific automizer properties for certain abelian subgroups, focusing on nilpotent and minimal non-nilpotent cases.

Findings

Characterization of groups with CGH = NGH for all relevant abelian subgroups

Complete classification of nilpotent groups with this property

Identification of minimal non-nilpotent groups with the property

Abstract

In this paper, we discuss about finite groups in which, CGH = NGH, for every abelian subgroup H of non prime power order. Also, we classify all such nilpotent and minimal non nilpotent groups.