Let G be a finite almost simple group with socle G0. A (nontrivial) factorization of G is an expression of the form G = HK, where the factors H and K are 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, Li and Xia 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 G0 containing H \ G0 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.
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Acknowledgments. This work was partially supported by NSFC grants 11231008 and 11771200. Burness thanks the Department of Mathematics at the Southern University of Science and Technology (SUSTech) for their generous hospitality during a research visit in April 2019. He also thanks Bob Guralnick for helpful discussions.
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- Almost simple groups
- Primitive groups
- Regular subgroups