Subgroups of direct products closely approximated by direct sums

TL;DR

This paper investigates conditions under which subgroups of direct products of groups can be approximated by direct sums, revealing that compactness and finiteness of component groups influence these approximation properties.

Contribution

It characterizes when controllability properties can be reversed in subgroups of direct products, especially highlighting the role of compactness and finite generation.

Findings

Reversal of the first controllability arrow occurs for compact subgroups.

The second arrow cannot be reversed even for compact subgroups.

All subgroups have reversible first arrow if all groups are finite.

Abstract

Let $I$ be an infinite set, $\{G_i:i\in I\}$ be a family of (topological) groups and $G=\prod_{i\in I} G_i$ be its direct product. For $J\subseteq I$, $p_{J}: G\to \prod_{j\in J} G_j$ denotes the projection. We say that a subgroup $H$ of $G$ is: (i) \emph{uniformly controllable} in $G$ provided that for every finite set $J\subseteq I$ there exists a finite set $K\subseteq I$ such that $p_{J}(H)=p_{J}(H\cap\bigoplus_{i\in K} G_i)$; (ii) \emph{controllable} in $G$ provided that $p_{J}(H)=p_{J}(H\cap\bigoplus_{i\in I} G_i)$ for every finite set $J\subseteq I$; (iii) \emph{weakly controllable} in $G$ if $H\cap \bigoplus_{i\in I} G_i$ is dense in $H$, when $G$ is equipped with the Tychonoff product topology. One easily proves that (i)$\to$(ii)$\to$(iii). We thoroughly investigate the question as to when these two arrows can be reversed. We prove that the first arrow can be reversed when $H$ is compact, but the second arrow cannot be reversed even when $H$ is compact. Both arrows can be reversed if all groups $G_i$ are finite. When $G_i=A$ for all $i\in I$, where $A$ is an abelian group, we show that the first arrow can be reversed for {\em all} subgroups $H$ of $G$ if and only if $A$ is finitely generated. Connections with coding theory are highlighted.