Random Generation of the Special Linear Group

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. \nAn 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 \\[ \n\\exp(-(2-o(1)) \\sqrt{n \\log n \\log q}). \\]