On finite permutation groups with the same orbits on unordered sets

Abstract

By JOHANNES SIEMONS and ASCHER WAGNER 1. Introduction. A permutation group G acting on a set f2 induces a permutation group on the unordered sets of k distinct points. If H is another permutation group on f2 we shall write H ~ G if H and G have the same orbits on the unordered sets of k points. Bercov and Hobby [2] have shown that for infinite groups H k G implies H ,,~ G if I < k. In [9] we have shown that this result is also true for finite groups, with the obviously necessary condition that k < 89 f2l- In [9] it is also shown that for finite groups H 2 G implies that H and G are either both primitive or both imprimitive with the same blocks of imprimi- tivity. If H and G have the same orbits on all subsets of ~2 we shall say that H and G are orbit equivalent and write H ~ G. Orbit equivalence for groups acting on quite arbitrary f2 has been considered by Betten [3]; the main results concern intransitive groups. In this paper we shall be concerned only with orbit equivalence for finite groups. In this case, of course, H ~ G if, and only if, H k G for all k. Suppose that H ~ G and that L is a permutation group on a set A. Then the direct products H x L and G x L, acting naturally on f2 u A, are orbit equivalent and intransi- tive. Also, if L is transitive on A the wreath products H 2, L and G 2, L, acting naturally on the direct sum of [A h copies of f2, are orbit equivalent and imprimitive. This suggests that the basic situation to investigate is when G, hence also H, is primitive on f2. Without loss of generality we may assume that H c G since H ~ G implies H ~ (H, G). Our main result is the following theorem. Theorem A. Let K be a finite primitive permutation group on a set f2. Let H c K and H ,~ K. Suppose there exists aprime r dividing the order of K but not the order of H. Then only the following possibilities exist: H K I g21 r 2-sets 3-sets 4-sets (i) ~3 5;3 3 2 (i 0 C 5 Dlo 5 2 5; 5 (iii) A~ (5) S s 5 3 10 (iv) A x (8) FA~ (S) 8 3 28 (V) A 1 (8) 23 \\PSL 3 (2) 8 3 28 (vi)