The exact number of r-regular elements in finite exceptional groups

TL;DR

This paper precisely calculates the number of r-regular elements in finite exceptional groups, revealing bounds on their proportions and demonstrating the existence of infinitely many groups with proportions arbitrarily close to a specific fraction.

Contribution

It provides exact counts of r-regular elements in finite exceptional groups and establishes bounds on their proportions, including the existence of groups with proportions near a specific value.

Findings

Proportion of r-regular elements is at least 3577/18432.

Infinitely many groups have proportions less than 3577/18432 + ε.

Exact counts for r-regular elements in finite exceptional groups.

Abstract

We calculate the precise number of r-regular elements in the finite exceptional groups. As a corollary we find that the proportion of r-regular elements is at least 3577/18432 and for all \epsilon>0, there are infinitely finite simple exceptional groups such that the proportion of r-regular elements is less than $3577/18432+\epsilon$ for some prime r.