Geometric Characterization of Property R
William Gordon Ritter
Abstract
Consider pairs of the form (G, N), with G a group and N G, as objects of a category . A morphism (G1, N1) (G2, N2) will be a group homomorphism f : G1 G2 such that f(N1) ⊂ N2. We introduce a functor Q : , 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.
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