On conjugacy classes and derived length

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 $2\eta(AB)$.