Subsets with restricted movement (Q5943068)

Let \(G\) be a permutation group acting on a (finite) set \(\Omega\) with no fixed-points in \(\Omega\). For a finite subset \(\Gamma\) of \(\Omega\) the movement under the action of \(G\) is defined by NEWLINE\[NEWLINE\text{move}(\Gamma)=\max_{g\in G}|\Gamma^g\setminus\Gamma|.NEWLINE\]NEWLINE The authors consider the case \(\max_{|\Gamma|=k}\text{move}(\Gamma)\leq m\) where \(m<k\leq|\Omega|\) and \(G\) is not a \(2\)-group.NEWLINENEWLINENEWLINEIn Theorem 1.1 and its corollaries several estimates of \(|\Omega|\) in terms of \(k\), \(m\) and the least odd prime dividing \(|G|\) are established.