Subgroups of direct products closely approximated by direct sums

Abstract

Abstract. Let I be an infinite set, {Gi : i ∈ I} be a family of (topological) groups and G = i∈I Gi be its direct product. For J ⊆ I, pJ : G → j∈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 ⊆ I there exists a finite set K ⊆ I such that pJ (H) = pJ (H ∩ i∈K Gi); (ii) controllable in G provided that pJ (H) = pJ (H ∩ i∈I Gi) for every finite set J ⊆ I; (iii) weakly controllable in G if H ∩ i∈I Gi is dense in H, when G is equipped with the Tychonoff product topology. One easily proves that (i)→(ii)→(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 Gi are finite. When Gi = A for all i ∈ 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. Connections with coding theory are highlighted.

Representation of Group Isomorphisms: The Compact Case

“…Another important example appears in coding theory, where the well known MacWilliams Equivalence Theorem asserts that, when G is a finite field and X and Y are finite sets, two codes (linear subspaces) A and B of G X and G Y , respectively, are equivalent when they are isometric for the Hamming metric (see [18,19,4]). This result has been generalized to convolutional codes in [12] and it also makes sense in other areas, as for example functional analysis and linear dynamical systems (cf [7,10,12,24,27]). The main motivation of this research has been to extend MacWilliams Equivalence Theorem to more general settings and explore the possible application of these methods to the study of convolutional codes or linear dynamical systems.…”

Weight-preserving isomorphisms between spaces of continuous functions: The scalar case

“…Thus, for each linear subspace of continuous functions considered along this paper, it is assumed: (1) for every x ∈ X there exists f ∈ A such that f (x) = 0. In coding theory, it is said that a convolutional code is controllable when any code sequence can be reached from the zero sequence in a finite interval (see [13,16,26,29]).…”

Homomorphic encoders of profinite abelian groups I

“…In this paper we investigate the structure of order controllable subgroups. It is known that each order controllable profinite abelian group is topologically isomorphic to a direct product of cyclic groups (see [8,15]). Here we improve this result and prove that under mild conditions an order controllable group G contains a set {g n : n ∈ N} that topologically generates G, and whose elements g n have all finite support.…”