On solvable factors of almost simple groups

2019/10/01 by Burness, Timothy C., Li, Cai Heng
#FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1910.00644

Abstract

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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