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

2024/10/03 by Beltrán, Antonio, Felipe, María José, Melchor, Carmen
#FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2410.02393

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∪ B or AB=A^-1 ∪ B, then G cannot be non-abelian simple. We demonstrate that, in fact, ⟨ A⟩ = ⟨ B⟩ is solvable, the elements of A and B are p-elements for some prime p, and ⟨ A⟩ 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.

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