On stabilizers in finite permutation groups

TL;DR

This paper investigates properties of stabilizers in finite permutation groups, establishing bounds on orbit lengths and derived lengths, and answering longstanding questions in group theory.

Contribution

It proves the existence of small orbit stabilizers in solvable groups and bounds the derived length of maximal subgroups in almost simple groups, advancing understanding of permutation group structure.

Findings

Existence of a set-stabilizer with orbits of length at most 6 in solvable groups

Maximal solvable subgroups of almost simple groups have derived length at most 10

In primitive groups with solvable stabilizer, some point stabilizer has bounded derived length

Abstract

Let $G$ be a permutation group on the finite set $\Omega$. We prove various results about partitions of $\Omega$ whose stabilizers have good properties. In particular, in every solvable permutation group there is a set-stabilizer whose orbits have length at most $6$, which is best possible and answers two questions of Babai. Every solvable maximal subgroup of any almost simple group has derived length at most $10$, which is best possible. In every primitive group with solvable stabilizer, there are two points whose stabilizer has derived length bounded by an absolute constant.