Derived length of stabilizers in finite permutation groups

Luca Sabatini

Published 2024-11-27Version 1

Let $G$ be a permutation group on the finite set $\Omega$. We study partitions of $\Omega$ whose stabilizers have bounded derived length, proving the following three theorems. In every solvable permutation group, there is a subset whose setwise stabilizer has derived length at most $3$, which is best possible. Every solvable maximal subgroup of any almost simple group has derived length bounded by an absolute constant. In every primitive group with solvable stabilizer, there are two points whose pointwise stabilizer has derived length bounded by an absolute constant.