Homomorphic encoders of profinite abelian groups

Let {Gi : i ∈ N} be a family of finite Abelian groups. We say that a subgroup G ≤ ∏ i∈N Gi is order controllable if for every i ∈ N there is ni ∈ N such that for each c ∈ G, there exists c1 ∈ G satisfying that c1|[1,i] = c|[1,i], supp(c1) ⊆ [1, ni], and order(c1) divides order(c|[1,ni]). In this paper we investigate the structure of order controllable subgroups. It is proved that every order controllable, profinite, abelian group contains a subset {gn : n ∈ N} that topologically generates the group and whose elements gn all have finite support. As a consequence, sufficient conditions are obtained that allow us to encode, by means of a topological group isomorphism, order controllable profinite abelian groups. Some applications of these results to group codes will appear subsequently [7].