September 28, 2025Open Access

On stabilizers in finite permutation groups

Key Points

In every solvable permutation group, there exists a set-stabilizer with orbits of length at most 6, providing a critical insight into group structure.
Every solvable maximal subgroup in an almost simple group has a derived length capped at 10, which showcases limitations in subgroup complexity.
Primitive groups with solvable stabilizers can guarantee that there are two points where the stabilizer's derived length is bounded by a fixed constant.
These results collectively enhance our understanding of the relationships among permutation groups, stabilizers, and their derived lengths.

Abstract

Abstract Let be a permutation group on the finite set . We prove various results about partitions of 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.

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