Subgroups of direct products closely approximated by direct sums

Let I be an infinite set, let {G(i) : i is an element of I} be a family of (topological) groups and let G = Pi(i is an element of J) Gi be its direct product. For J subset of I, p(J) : G Pi(i is an element of J) Gj denotes the projection. We say that a subgroup H of G is (i) uniformly controllable in G provided that for every finite set J subset of I there exists a finite set K subset of I such that pJ(H) = p(J)(H boolean AND circle plus (i is an element of K) G(i)), (ii) controllable in G provided that pJ(H) = p(J)(H boolean AND circle plus (i is an element of K) G(i)) for every finite set J subset of I, (iii) weakly controllable in G if H boolean AND circle plus (i is an element of I) G(i) is dense in H, when G is equipped with the Tychonoff product topology. One easily proves that ( i) double right arrow ( ii) double right arrow ( iii). We thoroughly investigate the question as to when these two arrows can be reversed. We prove that the first arrowcan 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 is an element of 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.