HewittMarczewskiPondiczery type theorem for abelian groups and Markov's potential density
Authors:
Dikran Dikranjan and Dmitri Shakhmatov
Journal:
Proc. Amer. Math. Soc. 138 (2010), 29792990
MSC (2010):
Primary 22A05; Secondary 20K99, 22C05, 54A25, 54B10, 54D65
Published electronically:
April 1, 2010
MathSciNet review:
2644909
Fulltext PDF
Abstract 
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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 .
Moreover, if , then the following item can be added to this list:  (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) torsionfree 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 .
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 7908577, Japan
Email:
dmitri@dpc.ehimeu.ac.jp
DOI:
http://dx.doi.org/10.1090/S0002993910103025
PII:
S 00029939(10)103025
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 P102920101 and J196430101 and by grant MTM200914409C0201
The second author was partially supported by GrantinAid 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.
