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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

The structure of $ \omega \sb{1}$-separable groups


Author: Paul C. Eklof
Journal: Trans. Amer. Math. Soc. 279 (1983), 497-523
MSC: Primary 03E35; Secondary 03E50, 20A15, 20K20
DOI: https://doi.org/10.1090/S0002-9947-1983-0709565-8
MathSciNet review: 709565
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Abstract: A classification theorem is proved for $ {\omega_1}$-separable $ {\omega_1}$-free abelian groups of cardinality $ {\omega_1}$ assuming Martin's Axiom $ ($MA$ )$ and $ {2^{\aleph_0}} > {\aleph_1}$. As a consequence, several structural results about direct sum decompositions of $ {\omega_1}$-separable groups are proved. These results are proved independent of ZFC, and, in addition, another structural property is proved undecidable in $ {\text{ZFC}} + {\text{MA}} + {2^{\aleph_0}} > {\aleph_1}$. The problem of classifying these groups in a model of $ {2^{\aleph_0}} = {\aleph_1}$ is also investigated.


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DOI: https://doi.org/10.1090/S0002-9947-1983-0709565-8
Article copyright: © Copyright 1983 American Mathematical Society