On transitive permutation groups with primitive subconstituents

Let G be a transitive permutation group on a set Omega such that, for omega is an element of Omega, the stabiliser G(omega) induces on each of its orb its in Omega\{omega} a primitive permutation group (possibly of degree 1). Let N be the normal closure of G(omega) in G. Then (Theorem 1) either N fac torises as N = G(omega)G(delta) for some omega, delta is an element of Omeg a, or all unfaithful G(omega)-orbits, if any exist, are infinite. This resu lt generalises a theorem of I. M. Isaacs which deals with the case where th ere is a finite upper bound on the lengths of the G(omega)-orbits. Several further results are proved about the structure of G as a permutation group, focussing in particular on the nature of certain G-invariant partitions of Omega.