A complete classification of solvable factors of almost simple groups

TL;DR

This paper fully classifies solvable factors in factorizations of almost simple groups, resolving a long-standing problem and providing new insights into group structures and permutation groups.

Contribution

It provides an explicit characterization of solvable factors in classical groups, completing the classification of such factors in almost simple groups.

Findings

Complete classification of solvable factors in classical groups

New characterization of one-dimensional transitive groups

Description of quasiprimitive permutation groups with solvable transitive subgroups

Abstract

We give an explicit characterization of solvable factors in factorizations of finite classical groups of Lie type. This completes the classification of solvable factors in factorizations of almost simple groups, finishing the program initiated in [Memoirs of the AMS, 279 (2022), no.~1375] and [Advances in Mathematics, 377 (2021), 107499]. In particular, it resolves the final remaining case in the long-standing problem of determining exact factorizations of almost simple groups. As a byproduct, we obtain a new characterization of one-dimensional transitive groups, offering further insights into their group structures. We also apply our classification to describe quasiprimitive permutation groups with a solvable transitive subgroup, leading to an interesting result that these subgroups are ``small''.