On the distribution of the density of maximal order elements in general linear groups

TL;DR

This paper investigates the distribution of maximal order elements in general linear groups over finite fields, providing expected values and distribution laws for their densities based on parameters like rank, characteristic, and extension degree.

Contribution

It introduces a novel analysis of the density distribution of maximal order elements in GL_n(q), including explicit expected values and distribution laws.

Findings

Expected density values computed for various parameters.

Distribution law established for the density of maximal order elements.

Insights into how group parameters influence element order distribution.

Abstract

In this paper we consider the density of maximal order elements in $\mathrm{GL}_n(q)$. Fixing any of the rank $n$ of the group, the characteristic $p$ or the degree $r$ of the extension of the underlying field $\mathbb{F}_q$ of size $q=p^r$, we compute the expected value of the said density and establish that it follows a distribution law.