Transitive permutation groups with elements of movement $m$ or $m-2$

Abstract

Let $G$ be a permutation group on a set $\Omega$ with no fixed points in $\Omega$ and let $m$ be a positive integer. If for each subset $\Gamma$ of $\Omega$ the size $|\Gamma^g\setminus\Gamma|$ is bounded, for $g\in G,$ we define the movement of $g$ as the $\max|\Gamma^g\setminus\Gamma|$ over all subsets $\Gamma$ of $\Omega,$ and the movement of $G$ is defined as the maximum of move$(g)$ over all non-identity elements of $g\in G.$ In this paper we classify all transitive permutation groups with bounded movement equal to $m$ that are not a $2$-group, but in which every non-identity element has movement $m$ or $m-2$.