On groups with no regular orbits on the set of subsets

Abstract

Let G be a permutation group on a finite set f2 of size n. Then G acts naturally on the set P (f2) of all subsets of f2. In this note we shall show that if G is primitive on f2 and A, $ G then in all except finitely many cases G has a regular orbit on P (O). We were led to consider this question by some recent work of Siemons and Wagner [5] and Inglis [4], where the following situation was considered. Given permutation groups G and H on the same finite set f2, write G ~ H if G and H have the same orbits on P (f2). It is shown in [4] and [5] that if G is primitive, A, ~ G and G ~ H then with a few explicitly known exceptions the same primes divide [GI and [HI. It is an immediate consequence of our result that under the same hypotheses with finitely many exceptions in fact G = H. (Indeed, with the weaker assumption that the longest orbits of G and H on P (f2) have the same length, we can conclude that [ G I = I HI with only finitely many exceptions.) Unlike the arguments in [4] and [5], ours require the classification of finite simple groups. We do however get a good deal of information even without the classification and can obtain our conclusion by elementary methods if G is 2-transitive on f2. The problem of determining pairs G and H of groups with G ~ H without the assumption of primitivity seems more difficult. Note, however, that if G ~ H and G is primitive then so is H: for G is primitive if and only if the only orbits of G on P (O) whose members partition f2 are the trivial ones. Similarly, but more easily, if G ~ H and G is transitive then so is H. We start with a preliminary well-known result. Recall that the minimal degree m of a permutation group G is the least number of points moved by a non-identity element of G, while the base size b is the least number of points of which the stabilizer is the identity.