Orbits of permutation groups on the power set (Q1592807)

Let \(G\) be a permutation group on a finite set \(\Omega\). The author investigates regular orbits of \(G\) on the set of all (ordered) partitions of \(\Omega\). If no alternating group \(A_n\) with \(n\geq 5\) is involved in \(G\), then \(G\) has a regular orbit consisting of partitions with 5 components (Corollary 6), and there exists a partition with 3 components which is fixed by no Sylow subgroup of \(G\) (Corollary 3). These results depend on work of \textit{A. Seress} [Bull. Lond. Math. Soc. 29, No.~6, 697-704 (1997; Zbl 0892.20002)], hence on the classification of finite simple groups.