Bases for quasisimple linear groups

Abstract

Let $V$ be a vector space of dimension $d$ over ${\mathbb{F}}_q$, a finite field of $q$ elements, and let $G\le GL\left(V\right)\cong {GL}_d\left(q\right)$ be a linear group. A base for $G$ is a set of vectors whose pointwise stabilizer in $G$ is trivial. We prove that if $G$ is a quasisimple group (i.e., $G$ is perfect and $G∕Z\left(G\right)$ is simple) acting irreducibly on $V$, then excluding two natural families, $G$ has a base of size at most 6. The two families consist of alternating groups ${Alt}_m$ acting on the natural module of dimension $d=m-1$ or $m-2$, and classical groups with natural module of dimension $d$ over subfields of ${\mathbb{F}}_q$.