Abstract

Let G be a transitive permutation group on a set Ω such that G is not a 2-group and let m be a positive integer. It was shown by the fourth author that if |�g\�| � m for every subset � of Ω and all g � G, then |Ω| � �2mp/(p-1)�, where p is the least odd prime dividing |G|. If p = 3 the upper bound for |Ω| is 3m, and the groups G attaining this bound were classified in the work of Gardiner, Mann, and the fourth author. Here we show that the groups G attaining the bound for p � 5 satisfy one of the following: (a) G := Zp � Z2a, |Ω| = p, m = (p-1)/2, where 2a|(p-1) for some a � 1; (b) G := K � P, |Ω| = 2sp, m = 2s-1(p-1), where 1 < 2s < p, K is a 2-group with p-orbits of length 2s, each element of K moves at most 2s(p-1) points of Ω, and P = Zp is fixed point free on Ω; (c) G is a p-group. All groups in case (a) are examples. In case (b), there exist examples for every p with s = 1. In case (c), where G is a p-group, we also prove that the exponent of G is bounded in terms of p only. Each transitive group of exponent p is an example, and it may be that these are the only examples in case (c). © 1999 Academic Press.