On solvable factors of almost simple groups

TL;DR

This paper establishes precise bounds and classifications for solvable factors in factorizations of almost simple groups, with applications to permutation groups containing nilpotent regular subgroups.

Contribution

It provides a sharp lower bound on the order of solvable factors and classifies all such factorizations, extending previous work on almost simple groups.

Findings

Determined the minimal order of solvable factors in almost simple groups.

Classified all factorizations with solvable factors.

Extended classical results on permutation groups with regular subgroups.

Abstract

Let $G$ be a finite almost simple group with socle $G_0$. 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 $G_0$ containing $H \cap G_0$ 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.