O'Brien, EamonnAn, JianbeiBaykalov, Anton A.2021-11-112021-11-112021https://hdl.handle.net/2292/57390Consider 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.Items in ResearchSpace are protected by copyright, with all rights reserved, unless otherwise indicated.Items in ResearchSpace are protected by copyright, with all rights reserved, unless otherwise indicated.https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmhttp://creativecommons.org/licenses/by-nc-sa/3.0/nz/Thesis2021-10-11Copyright: The authorhttp://purl.org/eprint/accessRights/OpenAccessQ112954779