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Because many systems of practical interest fall outside the scope of linear theory it is desirable to enlarge as much as possible the class of systems for which a complete structure theory is available. In this paper a class of finite state sequential systems evolving in groups is considered. The concepts of controllability, observability, minimality, realizability, and the isomorphism of minimal realizations are developed. Results which are analogous to - but differ in essential details from - those of linear system theory are derived. These results are potentially useful in such diverse areas as algorithmic design and algebraic coding.