On stabilizers in finite permutation groups

Abstract

Let $G$ be a permutation group on the finite set $\hat{I}\text{c◯}$. We prove various results about partitions of $\hat{I}\text{c◯}$ 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.

14 pages, to appear in Bull. London Math. Soc