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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Nonclosed pure subgroups of locally compact abelian groups


Author: Yuji Takahashi
Journal: Proc. Amer. Math. Soc. 110 (1990), 845-849
MSC: Primary 22B05
MathSciNet review: 1021905
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Abstract: We study the existence of many nonclosed pure subgroups of nondiscrete locally compact abelian groups. It is shown that every nondiscrete locally compact abelian group has uncountably many nonclosed pure subgroups. This in particular solves a question of Armacost. It is also shown that, if $ G$ is a nondiscrete locally compact abelian group and if either $ G$ is a compact group or the torsion part $ t\left( G \right)$ of $ G$ is nonopen, then $ G$ has $ {2^c}$ proper dense pure subgroups, where $ c$ denotes the power of the continuum. This in particular gives a partial answer to another question of Armacost.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1990-1021905-0
PII: S 0002-9939(1990)1021905-0
Keywords: Locally compact abelian groups, pure subgroups, nonclosed subgroups, dense subgroups
Article copyright: © Copyright 1990 American Mathematical Society