Homomorphic encoders of profinite abelian groups
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Let {G_i :i∈} be a family of finite Abelian groups. We say that a subgroup G≤∏_i∈G_i is order controllable if for every i∈ℕ there is n_i∈ℕ such that for each c∈ G, there exists c_1∈ G satisfying that c_1|[1,i]=c_|[1,i], supp (c_1)⊆ [1,n_i], and order(c_1) divides order(c_|[1,n_i]). In this paper we investigate the structure of order controllable subgroups. It is known that each order controllable profinite abelian group is topologically isomorphic to a direct product of cyclic groups (see <cit.>). Here we improve this result and prove that under mild conditions an order controllable group G contains a set {g_n : n∈} that topologically generates G, and whose elements g_n have all finite support. As a consequence, we obtain that if G is an order controllable, shift invariant, group code over an abelian group H, then G possesses a canonical generator set. Furthermore, our construction also yields that G is algebraically conjugate to a full group shift. Some connections to coding theory are also highlighted.
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