Toward a Group-Theoretic Proof of the Rearrangeability Theorem for Clos' Network

In this paper we continue the exploration begun in previous work 1 - 3 of the relationships between permutation groups and connecting networks t h a t are made of stages, frames, and cross-connect fields. Our results concern a well-known theoretical result of this area, the Slepian-Duguid theorem, which states t h a t Clos' three-stage network with square switches is rearrangeable, i.e., realizes any permutation. Since the permutations realizable by a stage form a special kind of subgroup, the theorem has been viewed in terms of group theory as a factorization of the symmetric group Snr of degree nr into a product of three subgroups or, alternatively, into a product of two mutually inverse double cosets. 3 We further illuminate this basic rearrangeability theorem by giving it as nearly group-theoretic a proof as we have been able to find. This proof starts from the known characterization 1 of the r?r-permutations 797 realizable by a Clos' three-stage network as a product G