  It is well known that the normaliser of a parabolic subgroup of a finite
Coxeter group is the semidirect product of the parabolic subgroup by the
stabiliser of a set of simple roots. We show that a similar result holds for
all finite unitary reflection groups: namely, the normaliser of a parabolic
subgroup is a semidirect product and in many, but not all, cases the complement
can be obtained as the stabiliser of a set of roots. In addition we record when
the complement acts as a reflection group on the space of fixed points of the
parabolic subgroup.
