Products of conjugacy classes and fixed point spaces
Robert Guralnick, Gunter Malle

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).
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