Subgroups of direct products closely approximated by direct sums

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.