Subgroups of direct products closely approximated by direct sums
Scholar | Other documents of the author: Ferrer González, María Vicenta; Hernández Muñoz, Salvador; Shakhmatov, Dmitry
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TitleSubgroups of direct products closely approximated by direct sums
Publisher versionhttps://www.degruyter.com/view/j/form.ahead-of-print/forum-2016-0047/www.degruyt ...
Let I be an infinite Π set, fGi : i 2 Ig be a family of (topological) groups and G = i∈I Gi be its direct product. For J I, pJ : G ! Π j∈J Gj denotes the projection. We say that a subgroup H of G is: (i) uniformly ... [+]
Let I be an infinite Π set, fGi : i 2 Ig be a family of (topological) groups and G = i∈I Gi be its direct product. For J I, pJ : G ! Π 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 2 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. [-]
Investigation project1) Generalitat Valenciana, grant code: PROMETEO II/2014/062; 2) Universitat Jaume I, grant P11B2015-77; 3) Grant-in-Aid for Scientific Research (C) No. 22540089 by the Japan Society for the Promotion of Science (JSPS).
Bibliographic citationFERRER GONZÁLEZ, María Vicenta; HERNÁNDEZ MUÑOZ, Salvador; SHAKHMATOV, Dmitri. Subgroups of direct products closely approximated by direct sums. Forum Mathematicum (2017), v. 19, issue 5
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