Random Generation of the Special Linear Group

TL;DR

This paper proves that pairs of elements in the special linear group (n,q) almost always generate the entire group as q^n grows, providing a new proof independent of the classification of finite simple groups.

Contribution

It offers a classification-free proof of a known result about generation in (n,q) and introduces an estimate for the average of the inverse order of elements in (n,q).

Findings

Proportion of generating pairs tends to 1 as q^n  n   q   n  inite simple groups.

New proof avoids classification of finite simple groups.

Estimates the average of 1/ord g over (n,q).

Abstract

It is well known that the proportion of pairs of elements of $\operatorname{SL}(n,q)$ which generate the group tends to $1$ as $q^n\to \infty$. This was proved by Kantor and Lubotzky using the classification of finite simple groups. We give a proof of this theorem which does not depend on the classification. An essential step in our proof is an estimate for the average of $1/\operatorname{ord} g$ when $g$ ranges over $\operatorname{GL}(n,q)$, which may be of independent interest. We prove that this average is \[ \exp(-(2-o(1)) \sqrt{n \log n \log q}). \]