Improvement on the Bound of Intransitive Permutation Groups with Bounded Movement

Abstract

Let $G$ be a permutation group on a set $\Omega$ with no fixed points in $\Omega$ and let $m$ be a positive integer. Then we define the movement of $G$ as, $m:=move(G):=sup_{\Gamma}\{|\Gamma^{g}\setminus\Gamma | | g\in G\}$. Let $p$ be a prime, $p\geq 5$, and let $move(G)=m$. We show that if $G$ is not a 2-group and $p$ is the least odd prime dividing $|G|$, then $n:=|\Omega|\leq 4m-p$. Moreover for an infinite family of groups the maximum bound $n=4m-p$ is attained.