Further solvable analogues of the Baer-Suzuki theorem and generation of nonsolvable groups

Let G be an almost simple group. We prove that if x is an element of G has prime order p >= 5, then there exists an involution y such that < x, y > is not solvable. Also, if x is an involution then there exist three conjugates of x that generate a nonsolvable group, unless x belongs to a short list of exceptions, which are described explicitly. We also prove that if x has order 6 or 9, then there exist two conjugates that generate a nonsolvable group. (c) 2012 Elsevier Inc. All rights reserved.