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.