Orbits of Permutation Groups

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 no
element of $G$ moves any subset of $\Omega$ by more than $m$
points (that is, if $|\Gamma^g \setminus \Gamma|\leq m$ for every
$\Gamma \subseteq \Omega$ and $g\in G$), and the lengths of all
orbits are not equal to $2$.
Then the number $t$ of $G$-orbits in $\Omega$ is at most $2m-2$.
Moreover, the groups attaining the maximum bound $t=2m-2$ will be classified. \vspace{.4cm}\\