Generation of finite classical groups by pairs of elements with large fixed point spaces

Abstract

We study `good elements' in finite $2n$-dimensional classical groups $G$: namely $t$ is a `good element' if $o(t)$ is divisible by a primitive prime divisor of $q^n-1$ for the relevant field order $q$, and $t$ fixes pointwise an $n$-space. The group ${\rm{SL}}_{2n}(q)$ contains such elements, and they are present in ${\rm{Su}}_{2n}(q), {\rm{Sp}}_{2n}(q), {\rm{So}}^\epsilon_{2n}(q)$, only if $n$ is odd, even, even, respectively. We prove that there is an absolute positive constant $c$ such that two random conjugates of $t$ generate $G$ with probability at least $c$, if $G\ne {\rm{Sp}}_{2n}(q)$ with $q$ even. In the exceptional case $G={\rm{Sp}}_{2n}(q)$ with $q$ even, two conjugates of $t$ never generate $G$: in this case we prove that two random conjugates of $t$ generate a subgroup ${\rm{SO}}^\epsilon_{2n}(q)$ with probability at least $c$. The results (proved for all field orders at least $4$) underpin analysis of new constructive recognition algorithms for classical groups in even characteristic, which succeed where methods utilising involution centralisers are not available.