Random Generation of the Special Linear Group

Abstract

It is well known that the proportion of pairs of elements of $\mathrm{SL}(n,q)$ 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 $1/\mathrm{ord}g$ when $g$ ranges over $\mathrm{GL}(n,q)$, which may be of independent interest. We prove that this average is \[ \exp(-(2-o(1)) \sqrt{n \log n \log q}). \]