On subgroups generated by small classes in finite groups

TL;DR

This paper investigates the structure of subgroups generated by small conjugacy classes in finite groups, focusing on their nilpotency properties and providing new conditional results.

Contribution

It offers new conditional theorems about the nilpotency class of subgroups generated by small conjugacy classes in finite groups.

Findings

Conditional bounds on the nilpotency class of M(G)

Results relating to the Fitting subgroup F(M(G))

Advances in understanding subgroup structure in finite groups

Abstract

Let $G$ be a finite group and $M(G)$ be the subgroup of $G$ generated by all non-central elements of $G$ that lie in the conjugacy classes of the smallest size. Recently several results have been proved regarding the nilpotency class of $M(G)$ and $F(M(G))$, where $F(M(G))$ denotes the Fitting subgroup of $M(G)$. We prove some conditional results regarding the nilpotency class of $M(G)$.