  Consider pairs of the form (G, N), with G a group and N \normal G, as objects
of a category \PG. A morphism (G_1, N_1) \To (G_2, N_2) will be a group
homomorphism f : G_1 \To G_2 such that f(N_1) \subset N_2. We introduce a
functor Q : \PG \To \Groups, which provides a geometric definition of Property
R, since it is most naturally visualized by means of a directed graph. We
compute these graphs for a number of finite groups of small order, and prove a
general characterization of the graphs which occur in this way.
