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

 

Hewitt-Marczewski-Pondiczery type theorem for abelian groups and Markov's potential density


Authors: Dikran Dikranjan and Dmitri Shakhmatov
Journal: Proc. Amer. Math. Soc. 138 (2010), 2979-2990
MSC (2010): Primary 22A05; Secondary 20K99, 22C05, 54A25, 54B10, 54D65
Published electronically: April 1, 2010
MathSciNet review: 2644909
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Abstract: For an uncountable cardinal $ \tau$ and a subset $ S$ of an abelian group $ G$, the following conditions are equivalent:

(i)
$ \vert\{ns:s\in S\}\vert\ge \tau$ for all integers $ n\ge 1$;
(ii)
there exists a group homomorphism $ \pi:G\to \mathbb{T}^{2^\tau}$ such that $ \pi(S)$ is dense in $ \mathbb{T}^{2^\tau}$.
Moreover, if $ \vert G\vert\le 2^{2^\tau}$, then the following item can be added to this list:
(iii)
there exists an isomorphism $ \pi:G\to G'$ between $ G$ and a subgroup $ G'$ of $ \mathbb{T}^{2^\tau}$ such that $ \pi(S)$ is dense in $ \mathbb{T}^{2^\tau}$.

We prove that the following conditions are equivalent for an uncountable subset $ S$ of an abelian group $ G$ that is either (almost) torsion-free or divisible:

(a)
$ S$ is $ \mathscr{T}$-dense in $ G$ for some Hausdorff group topology $ \mathscr{T}$ on $ G$;
(b)
$ S$ is $ \mathscr{T}$-dense in some precompact Hausdorff group topology $ \mathscr{T}$ on $ G$;
(c)
$ \vert\{ns:s\in S\}\vert\ge \min\left\{\tau:\vert G\vert\le 2^{2^\tau}\right\}$ for every integer $ n\ge 1$.
This partially resolves a question of Markov going back to 1946.


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Additional Information

Dikran Dikranjan
Affiliation: Università di Udine, Dipartimento di Matematica e Informatica, via delle Scienze, 206 - 33100 Udine, Italy
Email: dikran.dikranjan@dimi.uniud.it

Dmitri Shakhmatov
Affiliation: Division of Mathematics, Physics and Earth Sciences, Graduate School of Science and Engineering, Ehime University, Matsuyama 790-8577, Japan
Email: dmitri@dpc.ehime-u.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9939-10-10302-5
PII: S 0002-9939(10)10302-5
Keywords: Abelian group, monomorphism, homomorphism, potentially dense set, dense subset, precompact group
Received by editor(s): November 10, 2008
Received by editor(s) in revised form: November 24, 2009
Published electronically: April 1, 2010
Additional Notes: The first author was partially supported by SRA, grants P1-0292-0101 and J1-9643-0101 and by grant MTM2009-14409-C02-01
The second author was partially supported by Grant-in-Aid for Scientific Research (C) No. 19540092 of the Japan Society for the Promotion of Science (JSPS)
Communicated by: Alexander N. Dranishnikov
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.