Products and commutators of classes in algebraic groups

TL;DR

This paper classifies pairs of conjugacy classes in almost simple algebraic groups with finitely many product classes, generalizes a finite group theorem, and addresses a question on commutators in simple algebraic groups.

Contribution

It provides a classification of such pairs and establishes a connection to a generalization of the Baer--Suzuki theorem, answering a specific open question.

Findings

Identified all pairs with finitely many product classes

Linked these pairs to a generalized Baer--Suzuki theorem

Solved an open problem on commutators in simple algebraic groups

Abstract

We classify pairs of conjugacy classes in almost simple algebraic groups whose product consists of finitely many classes. This leads to several interesting families of examples which are related to a generalization of the Baer--Suzuki theorem for finite groups. We also answer a question of Pavel Shumyatsky on commutators of pairs of conjugacy classes in simple algebraic groups. It turns out that the resulting examples are exactly those for which the product also consists of only finitely many classes.