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

 

On the existence of $ \kappa $-free abelian groups


Author: Paul C. Eklof
Journal: Proc. Amer. Math. Soc. 47 (1975), 65-72
MSC: Primary 02K20; Secondary 20K35
MathSciNet review: 0379694
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Abstract: It is proved that if $ {\aleph _\alpha }$ is a regular cardinal such that there is an $ {\aleph _\alpha }$-free abelian group which is not $ {\aleph _{\alpha + 1}}$-free, then for every positive integer $ n$ there is an $ {\aleph _{\alpha + n}}$-free abelian group which is not $ {\aleph _{\alpha + n + 1}}$-free. A corollary is that for each positive integer $ n$ there is a group of cardinality $ {\aleph _n}$ which is $ {\aleph _n}$-free but not free. Some results on $ \kappa $-free abelian groups which involve notions from logic are also proved.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1975-0379694-0
PII: S 0002-9939(1975)0379694-0
Keywords: $ \kappa $-free abelian group, stationary set, equivalence in $ {L_{\infty \kappa }}$, axiom of constructibility, compact cardinal
Article copyright: © Copyright 1975 American Mathematical Society