Bounds on the number and sizes of conjugacy classes in finite Chevalley groups with Applications to Derangements

TL;DR

This paper provides explicit bounds on conjugacy classes in finite Chevalley groups, aiding various applications like zeta functions, random walks, and derangements, and solves a strong version of the Boston-Shalev conjecture for most cases.

Contribution

It introduces new bounds on conjugacy classes in finite Chevalley groups and proves a strong version of the Boston-Shalev conjecture for most primitive permutation representations.

Findings

Explicit upper bounds for conjugacy classes in Chevalley groups

Solution to a strong version of the Boston-Shalev conjecture for most cases

Applications to derangements and related problems

Abstract

We present explicit upper bounds for the number and size of conjugacy classes in finite Chevalley groups and their variations. These results have been used by many authors to study zeta functions associated to representations of finite simple groups, random walks on Chevalley groups, the final solution to the Ore conjecture about commutators in finite simple groups and other similar problems. In this paper, we solve a strong version of the Boston-Shalev conjecture on derangements in simple groups for most of the families of primitive permutation group representations of finite simple groups (the remaining cases are settled in two other papers of the authors and applications are given in a third).