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 ${\mathbf{C}}_G(A)=\{g\in G\mid x^g=xforallx\in A\}$. If $AB=D$ then ${\mathbf{C}}_G(D)/({\mathbf{C}}_G(A)\cap {\mathbf{C}}_G(B))$ is an abelian group. If, in addition, $G$ is supersolvable, then the derived length of ${\mathbf{C}}_G(D)/({\mathbf{C}}_G(A)\cap {\mathbf{C}}_G(B))$ is bounded above by $2\eta (AB)$.

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