Bases for quasisimple linear groups

Let [math] be a vector space of dimension [math] over [math] , a finite field of [math] elements, and let [math] be a linear group. A base for [math] is a set of vectors whose pointwise stabilizer in [math] is trivial. We prove that if [math] is a quasisimple group (i.e., [math] is perfect and [math] is simple) acting irreducibly on [math] , then excluding two natural families, [math] has a base of size at most 6. The two families consist of alternating groups [math] acting on the natural module of dimension [math] or [math] , and classical groups with natural module of dimension [math] over subfields of [math] .