Subgroups of direct products closely approximated by direct sums

Abstract

Abstract Let I be an infinite set, let { G i : i ∈ I } {\{G_{i}:i\in I\}} be a family of (topological) groups and let G = ∏ i ∈ I G i {G=\prod_{i\in I}G_{i}} be its direct product. For J ⊆ I {J\subseteq I} , p J : G → ∏ j ∈ J G j {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 ⊆ I {J\subseteq I} there exists a finite set K ⊆ I {K\subseteq I} such that p J ⁢ ( H ) = p J ⁢ ( H ∩ ⊕ i ∈ K G i ) {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 ∩ ⊕ i ∈ I G i ) {p_{J}(H)=p_{J}(H\cap\bigoplus_{i\in I}G_{i})} for every finite set J ⊆ I {J\subseteq I} , (iii) weakly controllable in G if H ∩ ⊕ i ∈ I G i {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 {G_{i}} are finite. When G i = A {G_{i}=A} for all i ∈ I {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.