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 | References | Similar Articles | Additional Information

Abstract: For an uncountable cardinal and a subset of an abelian group , the following conditions are equivalent:

- (i)
- for all integers ;
- (ii)
- there exists a group homomorphism such that is dense in .

- (iii)
- there exists an isomorphism between and a subgroup of such that is dense in .

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

- (a)
- is -dense in for some Hausdorff group topology on ;
- (b)
- is -dense in some precompact Hausdorff group topology on ;
- (c)
- for every integer .

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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:
https://doi.org/10.1090/S0002-9939-10-10302-5

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.