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By treating the scattering matrix as an operator, it is possible to relate the properties of cirulators to the cyclic substitutions of group theory and the oriented 1-circuits of topology. The body of knowledge made available by these two branches of mathematics is shown to yield precise definitions of circulator performance. Useful results in treating the symmetries, interconnections and cascade combinations of circulators are found by further application of group theory and topology. Practical application of the symmetry analysis is made to the design of circulators. It is shown that a necessary condition for circulator performance is that the structure under consideration possess no symmetry other than those specified in the circulator group. Thus, structures proposed for circulator development which do not meet this necessary requirement may be eliminated immediately with consequent savings in development time and expense.