Applications of Group Theory to Connecting Networks

Abstract

Group theory impinges on the combinatorial study of connecting networks in a natural way: the stages, frames, and cross-connect fields from which many existing networks are built provide simple permutations out of which desired, complex assignments are built by composition. Some of the consequences of this interpretation are explored in this paper. In the group-theoretic setting, the action and role of the stages and fields become transparent, and many questions and results regarding networks can be regarded as problems about cosets, subgroups, factorizations, etc. This approach is particularly useful for the study of rearrangeable networks made of stages of square switches; such a network is rearrangeable if and only if the symmetric group of appropriate degree can be factored into products of certain subgroups associated with the network. Or again, the original Slepian-Duguid rearrangeability theorem corresponds to factoring a symmetric group into a product of double cosets of subgroups generated by stages.