Simple groups admit Beauville structures

TL;DR

This paper proves that all finite simple groups except A_5 have unmixed Beauville structures, and explores related algebraic and generative properties of simple groups, providing new insights into their structure and representations.

Contribution

It confirms a conjecture that all finite simple groups except A_5 admit Beauville structures and investigates algebraic and generative properties of simple groups.

Findings

All finite simple groups except A_5 admit unmixed Beauville structures.

Finite simple groups contain conjugacy classes that generate the entire group.

Results depend on bounds for character values of regular semisimple elements.

Abstract

We answer a conjecture of Bauer, Catanese and Grunewald showing that all finite simple groups other than the alternating group of degree 5 admit unmixed Beauville structures. We also consider an analog of the result for simple algebraic groups which depends on some upper bounds for character values of regular semisimple elements in finite groups of Lie type and obtain definitive results about the variety of triples in semisimple regular classes with product 1. Finally, we prove that any finite simple group contains two conjugacy classes C,D such that any pair of elements in C x D generates the group.