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

Abstract

AbstractA 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 $$AB=A\cup B$$ A B = A ∪ B or $$AB=A^{-1} \cup B$$ A B = A - 1 ∪ B , then G cannot be a non-abelian simple group. We demonstrate that, in fact, $$\langle A\rangle = \langle B\rangle $$ ⟨ A ⟩ = ⟨ B ⟩ is solvable, the elements of A and B are p-elements for some prime p, and $$\langle A\rangle $$ ⟨ A ⟩ is p-nilpotent. Moreover, under the second assumption, it turns out that $$A=B$$ A = B . This research is done by appealing to recently developed techniques and results that are based on the Classification of Finite Simple Groups.