On conjugacy classes and derived length

TL;DR

This paper investigates the structure of conjugacy classes in finite groups, establishing bounds on the derived length of certain quotient groups in supersolvable groups based on the number of conjugacy classes involved.

Contribution

It introduces bounds on the derived length of specific quotient groups related to conjugacy classes in supersolvable groups, linking group structure to conjugacy class decompositions.

Findings

The quotient ${f C}_G(D)/({f C}_G(A)igcap{f C}_G(B))$ is abelian when $AB=D$.

In supersolvable groups, the derived length of this quotient is at most twice the number of conjugacy classes in the union.

The paper provides a new connection between conjugacy class decompositions and the derived length of associated groups.

Abstract

Let $G$ be a finite group and $A$, $B$ and $D$ be conjugacy classes of $ G$ with $D\subseteq AB=\{xy\mid x\in A, y\in B\}$. Denote by $\eta(AB)$ the number of distinct conjugacy classes such that $AB$ is the union of those. Set ${\bf C}_G(A)=\{g\in G\mid x^g=x {for all} x\in A\}$. If $AB=D$ then ${\bf C}_G(D)/({\bf C}_G(A)\cap{\bf C}_G(B))$ is an abelian group. If, in addition, $G$ is supersolvable, then the derived length of ${\bf C}_G(D)/({\bf C}_G(A)\cap{\bf C}_G(B))$ is bounded above by $2\eta(AB)$.