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Let $\{G_i :i\in\N\}$ be a family of finite Abelian groups. We say that asubgroup $G\leq \prod\limits_{i\in \N}G_i$ is \emph{order controllable} if forevery $i\in \mathbb{N}$ there is $n_i\in \mathbb{N}$ such that for each $c\inG$, there exists $c_1\in G$ satisfying that $c_{1|[1,i]}=c_{|[1,i]}$, $supp(c_1)\subseteq [1,n_i]$, and order$(c_1)$ divides order$(c_{|[1,n_i]})$. Inthis paper we investigate the structure of order controllable subgroups. It isproved that every order controllable, profinite, abelian group contains asubset $\{g_n : n\in\N\}$ that topologically generates the group and whoseelements $g_n$ all have finite support. As a consequence, sufficient conditionsare obtained that allow us to encode, by means of a topological groupisomorphism, order controllable profinite abelian groups. Some applications ofthese results to group codes will appear subsequently \cite{FH:2021}.

Homomorphic encoders of profinite abelian groups I