The maximal number of orbits of a permutation group with bounded movement
If \(G\) is a permutation group on \(\Omega\) with no fixed points, then \(G\) has bounded movement if there is a positive integer \(m\) such that for every \(g\in G\) and \(\Gamma\subseteq\Omega\) the cardinality \(|\Gamma^g\setminus\Gamma|\) is at most \(m\). If \(G\) has bounded movement, then the movement of \(G\) is the least such \(m\). It was shown by \textit{C. E. Praeger} that if \(G\) has movement \(m\) and no fixed points, then the number of \(G\)-orbits is at most \(2m-1\) [see J. Algebra 144, No. 2, 436-442 (1991; Zbl 0744.20004)]. In the present paper, there is a classification of the examples where the bound is attained. Apart from examples with \(|\Omega|=3\), these are elementary Abelian 2-groups of order \(2m\), and the point stabilisers of the \(G\)-orbits are the \(2m-1\) distinct subgroups of index 2. As pointed out by Avinoam Mann, it follows that if \(G\) has movement \(m\), and no fixed points on \(\Omega\), then \(|\Omega|=5m-2\) just when \(G\) is \(S_3\) or \(A_3\) (and \(|\Omega|=3\)). It was shown in the above paper of Praeger that \(|\Omega|\leq 5m-2\). The main theorem follows from another result, due to Peter M. Neumann and proved in this paper. Namely, if \(G\) has \(t\) orbits on \(\Omega\), \(\Gamma\subseteq\Omega\) and the maximum value of \(|\Gamma^g\setminus\Gamma|\) (for \(g\in G\)) is \(m>1\), and \(\Gamma\) has proper non-empty intersection with each \(G\)-orbit, then \(t\leq 2m-1\), and there is a description of the cases when equality holds.
scientific article; zbMATH DE number 1295431