Intersection of conjugate solvable subgroups in finite classical groups

Abstract

Consider the following problem stated by Vdovin (2010) in the "Kourovka notebook" (Problem 17.41): Let H be a solvable subgroup of a nite group G that has no nontrivial solvable normal subgroups. Do there always exist ve conjugates of H whose intersection is trivial? This problem is closely related to a conjecture by Babai, Goodman and Pyber (1997) about an upper bound for the index of a normal solvable subgroup in a nite group. The problem was reduced by Vdovin (2012) to the case when G is an almost simple group. Let G be an almost simple group with socle isomorphic to a simple linear, unitary or symplectic group, and assume that G contains neither graph nor graph- eld automorphisms of the socle. For all such groups G we provide a positive answer to Vdovin's problem.