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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

New criteria for freeness in abelian groups


Author: Paul Hill
Journal: Trans. Amer. Math. Soc. 182 (1973), 201-209
MSC: Primary 20K20
MathSciNet review: 0325805
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Abstract: A new criterion is established for an abelian group to be free. The criterion applies to an ascending chain of free subgroups. The result is used to construct groups that are almost free but not free. In particular, we construct examples that show that the class of free abelian groups is not definable in the logical language $ {L_{\infty \kappa }}$ if $ \kappa \leq {\aleph _2}$. In doing so, we take advantage of a recent theorem of P. Eklof.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1973-0325805-5
PII: S 0002-9947(1973)0325805-5
Keywords: Groups are abelian, free groups, $ {\aleph _1}$-freeness, $ {\aleph _2}$-freeness, smooth (ascending) chain, criteria for freeness, equivalence in $ {L_{\infty \kappa }}$
Article copyright: © Copyright 1973 American Mathematical Society