Any strongly controllable group system or group shift or any linear block code is isomorphic to a generator group
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Consider any sequence of finite groups A^t, where t takes values in an integer index set 𝐙. A group system A is a set of sequences with components in A^t that forms a group under componentwise addition in A^t, for each t∈𝐙. As shown previously, any strongly controllable complete group system A can be decomposed into generators. We study permutations of the generators when sequences in the group system are multiplied. We show that any strongly controllable complete group system A is isomorphic to a generator group (𝒰,∘). The set 𝒰 is a set of tensors, a double Cartesian product space of sets G_k^t, with indices k, for 0≤ k≤ℓ, and time t, for t∈𝐙. G_k^t is a set of unique generator labels for the generators in A with nontrivial span for the time interval [t,t+k]. We show the generator group contains a unique elementary system, an infinite collection of elementary groups, one for each k and t, defined on small subsets of 𝒰, in the shape of triangles, which form a tile like structure over 𝒰. There is a homomorphism from each elementary group to any elementary group defined on smaller tiles of the former group. The group system A may be constructed from either the generator group or elementary system. These results have application to linear block codes, any algebraic system that contains a linear block code, group shifts, and harmonic theory in mathematics, and systems theory, coding theory, control theory, and related fields in engineering.
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