On the uniform spread of almost simple linear groups

Abstract

Let $G$ be a finite group, and let $k$ be a nonnegative integer. We say that $G$ has uniform spread $k$ if there exists a fixed conjugacy class $C$ in $G$ with the property that for any $k$ nontrivial elements $x_1,\dots ,x_k$ in $G$ there exists $y\in C$ such that $G=\langle x_i,y\rangle$ for all $i$. Further, the exact uniform spread of $G$, denoted by $u\left(G\right)$, is the largest $k$ such that $G$ has the uniform spread $k$ property. By a theorem of Breuer, Guralnick, and Kantor, $u\left(G\right)\ge 2$ for every finite simple group $G$. Here we consider the uniform spread of almost simple linear groups. Our main theorem states that if $G=\langle {PSL}_n\left(q\right),g\rangle$ is almost simple, then $u\left(G\right)\ge 2$ (unless $G\cong S_6$), and we determine precisely when $u\left(G\right)$ tends to infinity as $\left|G\right|$ tends to infinity.