Homomorphic encoders of profinite abelian groups I

Let ${Gi :i\in\N}$ be a family of finite Abelian groups. We say that a subgroup $G\leq \prod\limits{i\in \N}Gi$ is \emph{order controllable} if for every $i\in \mathbb{N}$ there is $ni\in \mathbb{N}$ such that for each $c\in G$, there exists $c1\in G$ satisfying that $c{1|[1,i]}=c{|[1,i]}$, $supp (c1)\subseteq [1,ni]$, 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 ${gn : n\in\N}$ 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 \cite{FH:2021}.