On solvable factors of almost simple groups

Let Gbe a finite almost simple group with socle G(0). A (nontrivial) factorization of Gis an expression of the form G = HK, where the factors Hand Kare core-free subgroups. There is an extensive literature on factorizations of almost simple groups, with important applications in permutation group theory and algebraic graph theory. In a recent paper, Xia and the second-named author describe the factorizations of almost simple groups with a solvable factor H. Several infinite families arise in the context of classical groups and in each case a solvable subgroup of G(0) containing H boolean OR G(0) is identified. Building on this earlier work, in this paper we compute a sharp lower bound on the order of a solvable factor of every almost simple group and we determine the exact factorizations with a solvable factor. As an application, we describe the finite primitive permutation groups with a nilpotent regular subgroup, extending classical results of Burnside and Schur on cyclic regular subgroups, and more recent work of Li in the abelian case. (C) 2020 Elsevier Inc. All rights reserved.