Let G be a transitive permutation group on a set Ω
such that, for ω∈Ω, the stabiliser Gω induces on
each of its orbits in Ω\{ω} a primitive permutation group
(possibly of degree 1). Let N be the normal closure
of Gω in G. Then (Theorem 1) either N
factorises as N=GωGδ for some
ω, δ∈Ω, or all unfaithful Gω-orbits,
if any exist, are infinite. This result generalises a theorem of I. M.
Isaacs which deals with the case where
there is a finite upper bound on the lengths of the Gω-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 Ω.