Simple groups admit Beauville structures

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.

30 pages, in the second version, some results are improved and in particular we prove an irreducibility for a certain variety

References in corpus (4)

• Beauville surfaces and finite simple groups
• Symmetric powers and a problem of Kollar and Larsen
• Generation of finite simple groups with an application to groups acting on Beauville surfaces
• Beauville surfaces and finite groups

Cited by in corpus (24)

• Beauville surfaces and finite simple groups
• Series of $p$-groups with Beauville structure
• Generation of finite simple groups with an application to groups acting on Beauville surfaces
• Invariable generation of finite classical groups
• Generating sets of finite groups
• Invariable generation and the chebotarev invariant of a finite group
• Automorphism groups of Beauville surfaces
• Some Exceptional Beauville Structures
• Deformation theory and finite simple quotients of triangle groups I
• Finite groups satisfying the independence property
• Almost all alternating groups are invariably generated by two elements of prime order
• Crystalline Chebotarëv density theorems
• On the asymptotic behaviour of the number of Beauville and non-Beauville $p$-groups
• Beauville $p$-groups of wild type and groups of maximal class
• On invariable generation of alternating groups by elements of prime and prime power order
• Doubly Hurwitz Beauville groups
• Beauville structures in $p$-central quotients
• Normal coverings of linear groups
• Metabelian thin Beauville $p$-groups
• Invariable Generation of Infinite Groups
• Character bounds for regular semisimple elements and asymptotic results on Thompson's conjecture
• Normal coverings and pairwise generation of finite alternating and symmetric groups
• Coxeter groups as Beauville groups
• Primitive permutation groups and derangements of prime power order