AbstractLet I be an infinite set, let {\{G_{i}:i\in I\}} be a family of (topological) groups and let
{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)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)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)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) {\Rightarrow} (ii) {\Rightarrow} (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 all subgroups H of G if and only if
A is finitely generated.
We also describe compact groups topologically isomorphic to a direct product of countably many cyclic groups.
Connections with coding theory are highlighted.
Subgroups of finite direct sums of valuated cyclic groups
