Transitive Permutation Groups with Bounded Movement Having Maximal Degree

LetGbe a transitive permutation group on a set Ω such thatGis not a 2-group and letmbe a positive integer. It was shown by the fourth author that if Γg\Γ ≤ mfor every subset Γ of Ω and allg G, then Ω ≤ 2mp/(p − 1) , wherepis the least odd prime dividing G. Ifp = 3 the upper bound for Ω is 3m, and the groupsGattaining this bound were classified in the work of Gardiner, Mann, and the fourth author. Here we show that the groupsGattaining the bound forp ≥ 5 satisfy one of the following: (a)G Zp Z2a, Ω = p,m = (p − 1)/2, where 2a(p − 1) for somea ≥ 1; (b)G K P, Ω = 2sp,m = 2s − 1(p − 1), where 1 < 2s < p,Kis a 2-group withp-orbits of length 2s, each element ofKmoves at most 2s(p − 1) points of Ω, andP = Zpis fixed point free on Ω; (c)Gis ap-group. All groups in case (a) are examples. In case (b), there exist examples for everypwiths = 1. In case (c), whereGis ap-group, we also prove that the exponent ofGis bounded in terms ofponly. Each transitive group of exponentpis an example, and it may be that these are the only examples in case (c).