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Algebraic structure of small countably compact Abelian groups

Algebraic structure of small countably compact Abelian groups Abstract. Under Martin's Axiom, we completely characterize the algebraic structure of Abelian groups of the size c that admit a countably compact Hausdor¤ group topology. It turns out that, in the torsion case, these are exactly the groups that admit a pseudocompact Hausdor¤ group topology, but this is no more valid for non-torsion Abelian groups. The algebraic constraints for the existence of a countably compact Hausdor¤ group topology on an Abelian group G of size c are relatively simple: for every integer n, the subgroup G½n ¼ fx A G : nx ¼ 0g of G has to be finite or satisfy jG½nj ¼ c. The same has to hold true for the subgroup dG½n, where d is any divisor of n. In addition, if G is non-torsion, then the free rank rðGÞ of G must be equal to c. 2000 Mathematics Subject Classification: 22A05, 54H11; 54A25, 54D30, 54A35. 1 Introduction Most of the theory of Abelian groups is built on the study of appropriate cardinal invariants of the groups. Similarly, the role of cardinal invariants of the topological spaces is fundamental in general topology. Therefore, it is not an exaggeration to say that cardinal invariants are the http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

Algebraic structure of small countably compact Abelian groups

Forum Mathematicum , Volume 15 (6) – Oct 1, 2003

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References (28)

Publisher
de Gruyter
Copyright
Copyright © 2003 by Walter de Gruyter GmbH & Co. KG
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/form.2003.041
Publisher site
See Article on Publisher Site

Abstract

Abstract. Under Martin's Axiom, we completely characterize the algebraic structure of Abelian groups of the size c that admit a countably compact Hausdor¤ group topology. It turns out that, in the torsion case, these are exactly the groups that admit a pseudocompact Hausdor¤ group topology, but this is no more valid for non-torsion Abelian groups. The algebraic constraints for the existence of a countably compact Hausdor¤ group topology on an Abelian group G of size c are relatively simple: for every integer n, the subgroup G½n ¼ fx A G : nx ¼ 0g of G has to be finite or satisfy jG½nj ¼ c. The same has to hold true for the subgroup dG½n, where d is any divisor of n. In addition, if G is non-torsion, then the free rank rðGÞ of G must be equal to c. 2000 Mathematics Subject Classification: 22A05, 54H11; 54A25, 54D30, 54A35. 1 Introduction Most of the theory of Abelian groups is built on the study of appropriate cardinal invariants of the groups. Similarly, the role of cardinal invariants of the topological spaces is fundamental in general topology. Therefore, it is not an exaggeration to say that cardinal invariants are the

Journal

Forum Mathematicumde Gruyter

Published: Oct 1, 2003

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