On finite permutation groups with the same orbits on unordered sets (Q1068929)

Let G and H be permutation groups acting on a set \(\Omega\). Then \(H\approx^{k}G\) iff H and G have the same orbits on the unordered sets of k points. G and H are said to be orbit equivalent (H\(\approx G)\) iff \(H\approx^{k}G\) for all k. The main result of the present paper is the following Theorem A: Let K be a finite primitive permutation group on a set \(\Omega\). Let \(H<K\) and \(H\approx K\). Suppose there exists a prime r dividing the order of K but not the order of H. Then \(| \Omega |\) belongs to \(\{\) 3,5,8,9\(\}\) and the possibilities for H and K are explicitly stated. A somehow stronger version of the result, Theorem B, is proved using the classification of 3-transitive groups. Remark. The concept of orbit equivalence should be compared with \textit{H. Wielandt}'s concept of k- equivalence and k-closure, see [Permutation groups through invariant relations and invariant functions; Lecture notes, Ohio State Univ. (1969)]. Hence Theorem A corresponds in some way to a well known theorem of \textit{R. A. Beaumont} and \textit{R. P. Peterson} [Can. J. Math. 7, 35-42 (1955; Zbl 0064.025)].