Products of conjugacy classes and fixed point spaces

TL;DR

This paper investigates products of conjugacy classes in finite simple groups, establishing uniform generating triples, resolving a 1966 conjecture on fixed point spaces, and demonstrating that every element can be expressed as a product of two rth powers.

Contribution

It provides new results on conjugacy class products, including a uniform generating triple and solutions to longstanding conjectures in finite simple groups.

Findings

Existence of a uniform generating triple in finite simple groups

Resolution of a 1966 conjecture on fixed point spaces in linear groups

Every element can be written as a product of two rth powers

Abstract

We prove several results on products of conjugacy classes in finite simple groups. The first result is that there always exists a uniform generating triple. This result and other ideas are used to solve a 1966 conjecture of Peter Neumann about the existence of elements in an irreducible linear group with small fixed space. We also show that there always exist two conjugacy classes in a finite non-abelian simple group whose product contains every nontrivial element of the group. We use this to show that every element in a non-abelian finite simple group can be written as a product of two rth powers for any prime power r (in particular, a product of two squares).