Invariable generation of finite classical groups.

A subset of a group invariably generates the group if it generates even when we replace the elements by any of their conjugates. In a 2016 paper, Pemantle, Peres and Rivin show that the probability that four randomly selected elements invariably generate S n is bounded away from zero by an absolute constant for all n. Subsequently, Eberhard, Ford and Green have shown that the probability that three randomly selected elements invariably generate S n tends to zero as n → ∞. In this paper, we prove an analogous result for the finite classical groups. More precisely, let G r (q) be a finite classical group of rank r over F q. We show that for q large enough, the probability that four randomly selected elements invariably generate G r (q) is bounded away from zero by an absolute constant for all r , and for three elements the probability tends to zero as q → ∞ and r → ∞. We use the fact that most elements in G r (q) are separable and the well-known correspondence between classes of maximal tori containing separable elements in classical groups and conjugacy classes in their Weyl groups. [ABSTRACT FROM AUTHOR]