An Arad and Fisman’s Theorem on Products of Conjugacy Classes Revisited

Abstract

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