Transitive permutation groups with bounded movement having maximal degree

Let G be a transitive permutation group on a set Omega such that G is not a 2-group and let m be a positive integer. It was shown by the fourth author that if \Gamma(g)\Gamma\ less than or equal to m for every subset Gamma of Omega and all g is an element of G, then \Omega\ less than or equal to rig ht perpendicular 2mp/(p - 1) left perpendicular, where p is the least odd p rime dividing \G\. If p = 3 the upper bound for \Omega\ is 3m, and the grou ps G attaining this bound were classified in the work of Gardiner, Mann, an d the fourth author. Here we show that the groups G attaining the bound for p greater than or equal to 5 satisfy one of the following: (a) G := Z(p) x Z(2a), \Omega\ = p, m = (p - 1)/2, where 2(a)\(p - 1) for some a greater t han or equal to 1; (b) G := K x P, \Omega\ = 2(s) p, m = 2(s-1) (p - 1), wh ere 1 < 2(s) < p, K is a 2-group with p-orbits of length 2(s), each element of K moves at most 2(s) (p - 1) points of Omega, and P = Z(p) is fixed poi nt free on Omega; (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 o f p only. Each transitive group of exponent p is an example, and it may be that these are the only examples in case (c). (C) 1999 Academic Press.