PRIMITIVE PERMUTATION GROUPS WITH A DOUBLY TRANSITIVE SUBCONSTITUENT

Abstract Let G be a primitive permutation group on a finite set fi. We investigate the subconstitutentsof G, that is the permutation groups induced by a point stabilizer on its orbits in 0, in the caseswhere G has a diagonal action or a product action on fl. In particular we show in these casesthat no subconstituent is doubly transitive. Thus if G has a doubly transitive subconstituentwe show that G has a unique minimal normal subgrou TpV and eithe Tr V is a nonabelian simplegroup or N acts regularly on Q: we investigate further the case where N is regular on fl. 1980 Mathematics subject classification Soc.): 2 (Amer.0 B 15, 05 C Math. 25. Finite primitive permutation groups with a doubly transitive subconstituent werefirst studied by W. A. Manning (see [12, 17.7]). His results were generalized byP. J. Cameron ([2] and see also [4, 8, 9]). The analogues of these groups in thearea of symmetric graphs, namely, 2-arc transitive graphs, have also received agreat deal of attention in the literature. In this paper we show that these groupshave a unique minimal normal subgroup which either is a nonabelian simplegroup or is regular. We begin by studying the nature of the subconstituents ofa primitive group G with a diagonal or a product action: we show in particular