Products of conjugacy classes in finite and algebraic simple groups

TL;DR

This paper proves the Arad-Herzog conjecture for various finite simple groups, showing that the product of two nontrivial conjugacy classes generally does not form a single conjugacy class, and extends results to algebraic groups and p-elements.

Contribution

It establishes the Arad-Herzog conjecture for finite simple groups and algebraic groups, and classifies dense products of centralizers in simple algebraic groups.

Findings

Product of two nontrivial conjugacy classes is not a conjugacy class in many cases.

Almost all products of two conjugacy classes in simple algebraic groups contain infinitely many classes.

Complete classification of pairs of centralizers with dense product in simple algebraic groups.

Abstract

We prove the Arad-Herzog conjecture for various families of finite simple groups- if A and B are nontrivial conjugacy classes, then AB is not a conjugacy class. We also prove that if G is a finite simple group of Lie type and A and B are nontrivial conjugacy classes, either both semisimple or both unipotent, then AB is not a conjugacy class. We also prove a strong version of the Arad-Herzog conjecture for simple algebraic groups and in particular show that almost always the product of two conjugacy classes in a simple algebraic group consists of infinitely many conjugacy classes. As a consequence we obtain a complete classification of pairs of centralizers in a simple algebraic group which have dense product. In particular, there are no dense double cosets of the centralizer of a noncentral element. This result has been used by Prasad in considering Tits systems for psuedoreductive groups. Our final result is a generalization of the Baer-Suzuki theorem for p-elements with p a prime at least 5.