Information, Communication, and Group Theory

Information theory, as it is known today, resulted from the confluence of two very different roots that had their origins in the first half of the 20th century. On the one hand, information theory originated from electrical engineers such as Hartley, Nyquist, and Shannon [49, 86, 104], who worked on the analysis of systems and strategies to communicate messages from one location to another. On the other hand, mathematicians such as de Bruijn, Cramér, Fisher, Kullbach, and Rao were inventing ideas in probability and statistics that have direct relevance to the study of information transmission. In this chapter the “communications” aspect of information theory is emphasized, whereas in Chapter 3 the “probability and statistics” side was reviewed. In recent years, the theory of finite groups has been connected with equalities in information theory. Lie groups enter as symmetry operations associated with continuous physical models of information transmission such as the linear telegraph equation and nonlinear soliton equations. Lie groups also appear as a domain in which stochastic trajectories evolve in the analysis of noise in optical communication systems that transmit information over fiber optic cables. In addition, some of the basic concepts and definitions in the theory of communication have interesting properties that are enriched by merging them with concepts from group theory. Some of this recent work will be explored here.