Certain groups with bounded movement having the maximal number of orbits (Q1849090)

Let \(G\) be an intransitive permutation group acting on a set \(\Omega\) such that \(G\) has no fixed points in \(\Omega\). Let \(m\) be a positive integer. If \(G\) has bounded movement \(m\) then by \textit{C. E. Praeger} [J. Algebra 144, No. 2, 436-442 (1991; Zbl 0744.20004)] \(|\Omega|\) is finite. It is shown in Theorem 1 of the paper that, if \(p\) (\(\geq 3\)) is the least odd prime dividing the order of \(G\), then the number of \(G\)-orbits is at most \(2m-(p-1)\). In Theorem 2, all groups \(G\) of bounded movement equal to \(m\) having just \(2m-(p-1)\) orbits in \(\Omega\) are classified under the assumption that \(p\) (\(\geq 5\)) is the least odd prime dividing the order of \(G\). In any such case \(|G|=2^ap^b\) for suitable integers \(a\) and \(b\) and all orbits of \(G\) have length \(2\) except one orbit of length \(p\).