Subgroups of direct products closely approximated by direct sums
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Show full item recordcomunitat-uji-handle:10234/9
comunitat-uji-handle2:10234/7037
comunitat-uji-handle3:10234/8635
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Title
Subgroups of direct products closely approximated by direct sumsDate
2017-09Publisher
De GruyterBibliographic citation
FERRER GONZÁLEZ, María Vicenta; HERNÁNDEZ MUÑOZ, Salvador; SHAKHMATOV, Dmitri. Subgroups of direct products closely approximated by direct sums. Forum Mathematicum (2017), v. 19, issue 5Type
info:eu-repo/semantics/articlePublisher version
https://www.degruyter.com/view/j/form.ahead-of-print/forum-2016-0047/www.degruyt ...Version
info:eu-repo/semantics/acceptedVersionSubject
Abstract
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. [-]
Is part of
Forum Mathematicum (2017), v. 29, issue 5Investigation project
1) 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).Rights
http://rightsstatements.org/vocab/CNE/1.0/
info:eu-repo/semantics/openAccess
info:eu-repo/semantics/openAccess
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