Homomorphic encoders of profinite abelian groups I

TL;DR

This paper studies the structure of order controllable subgroups of profinite abelian groups, showing they can be generated by elements with finite support and providing conditions for their topological encoding, with applications to group codes.

Contribution

It introduces the concept of order controllability for subgroups of profinite abelian groups and establishes their structural properties and encoding conditions.

Findings

Order controllable profinite abelian groups are topologically generated by elements with finite support.

Sufficient conditions for encoding these groups via topological isomorphisms.

Applications to group coding theory are discussed.

Abstract

Let $\{G_i :i\in\N\}$ be a family of finite Abelian groups. We say that a subgroup $G\leq \prod\limits_{i\in \N}G_i$ is \emph{order controllable} if for every $i\in \mathbb{N}$ there is $n_i\in \mathbb{N}$ such that for each $c\in G$, 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$(c_{|[1,n_i]})$. In this paper we investigate the structure of order controllable subgroups. It is proved that every order controllable, profinite, abelian group contains a subset $\{g_n : n\in\N\}$ that topologically generates the group and whose elements $g_n$ 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}.