On the Sylow subgroups of a doubly transitive permutation group

Let G toe a 2-transitive permutation group of a set !! of npoints and let P be a Sylow p-subgroup of G where p is aprime dividing \G\ . If we restrict the lengths of the orbitsof P , can we correspondingly restrict the order of P ? In theprevious two papers of this series we were concerned with thecase in which all P-orbits have length at most p ; in thesecond paper we looked at Sylow p-subgroups of a two pointstabiliser. We showed that either P had order p , orG > A , G = PSL(2, 5) with p = 2 , or G = M of degree 12with p = 3 . In this paper we assume that P has a subgroup Qof index p and all orbits of Q have length at most p . We2conclude that either P has order at most p , or the groupsare known; namely PSL(3, p) S G5 PGL(3, p) ,ASL(2, p) £ G 5 AGL(2, p) , G = PrL(2, 8) with p = 3 ,G = M with p = 3 , G = PGL(2, 5) with p = 2 , or G > A2with 3p - n < 2p ; all in their natural representations.Let G be a doubly transitive permutation group on a set ft of npoints and let P be a Sylow p-subgroup of G where p is a primedividing \G\ . The previous two papers [9, JO] were concerned with thesituation in which P has no orbit of length greater than p . We showedessentially that either G contains the alternating group or P has orderp . The general problem is the following:Received 21 May 1975.