Fundamental groups of a strongly controllable group system or group code (including group shift and linear block code)

A finite group G defines a signature group if and only if there is a normal chain of G and a complete set of coset representatives of the normal chain that can be arranged in a triangular form, where representatives in the same row can be put in 1-1 correspondence so that certain sets of subtriangles of representatives form isomorphic groups, when subtriangle multiplication is induced by multiplication in G. We show that any strongly controllable time invariant group system (or group shift or group code) can be associated with a signature group. Conversely given a signature group, we can construct a strongly controllable time invariant group system. These results show that to some extent the study of strongly controllable time invariant group systems can be reduced to the study of signature groups. For a general strongly controllable group system which may vary in time, we show there is a signature sequence of groups with similar properties as a signature group. The general strongly controllable group system can be restricted to a finite interval, in which case it becomes a block group system. We describe the additive structure of the generator vectors which comprise the block group system. This new result applies to any mathematical structure defined on a finite number of coordinates which is closed under group addition in each coordinate, such as any linear block code over a group, field, vector space, ring, or module. Finally we give algorithms to construct any strongly controllable group system or its time invariant version or block version.