An Arad and Fisman's theorem on products of conjugacy classes revisited

TL;DR

This paper revisits a theorem about conjugacy classes in finite groups, showing that certain product conditions imply the involved classes generate a solvable, p-nilpotent subgroup, with specific cases leading to equality of classes.

Contribution

It refines the original theorem by establishing solvability, p-nilpotency, and element properties under product conditions, using recent classification techniques.

Findings

Generated subgroup is solvable and p-nilpotent.

Elements are p-elements for some prime p.

Under the second condition, conjugacy classes are equal.

Abstract

A theorem of Z. Arad and E. Fisman establishes that if $A$ and $B$ are two conjugacy classes of a finite group $G$ such that either $AB=A\cup B$ or $AB=A^{-1} \cup B$, then $G$ cannot be non-abelian simple. We demonstrate that, in fact, $\langle A\rangle = \langle B\rangle$ is solvable, the elements of $A$ and $B$ are $p$-elements for some prime $p$, and $\langle A\rangle $ is $p$-nilpotent. Moreover, under the second assumption, it turns out that $A=B$ and this is the only possible case. This research is done by appealing to recently developed techniques and results that are based on the Classification of Finite Simple Groups.