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Transactions of the American Mathematical Society

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New criteria for freeness in abelian groups. II

Author: Paul Hill
Journal: Trans. Amer. Math. Soc. 196 (1974), 191-201
MSC: Primary 20K20
MathSciNet review: 0352294
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Abstract: A new criterion is established for an abelian group to be free. The criterion is in terms of an ascending chain of free subgroups and is dependent upon a new class of torsion-free groups. The result leads to the construction, for each positive integer n, of a group $ {G_n}$ of cardinality $ {\aleph _n}$ that is not free but is $ {\aleph _n}$-free. A conjecture in infinitary logic concerning free abelian groups is also verified.

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  • [1] P. Eklof, Infinitary equivalence of abelian groups: with counterexamples to Scott's theorem in uncountable cardinals (preprint). MR 0354349 (50:6829)
  • [2] S. Feferman, Infinitary properties, local functors, and systems of ordinal functions, Conference in Mathematical Logic (London, 1970), Springer-Verlag, Berlin, 1972. MR 0360196 (50:12646)
  • [3] L. Fuchs, Infinite abelian groups. Vol. I, Pure and Appl. Math., vol. 36, Academic Press, New York, 1970. MR 41 #333. MR 0255673 (41:333)
  • [4] P. Griffith, $ {\aleph _n}$-free abelian groups, Aarhus University preprint series, 1971/72. MR 0325804 (48:4150)
  • [5] P. Hill, On the splitting of modules and abelian groups, Canad. J. Math. 26 (1974), 68-77. MR 0338217 (49:2983)
  • [6] -, New criteria for freeness in abelian groups, Trans. Amer. Math. Soc. 182 (1973), 201-209. MR 0325805 (48:4151)

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Keywords: Smooth chain of free groups, $ {\aleph _n}$-free, $ {\beth _n}$-free groups, non-free $ {\aleph _n}$-free groups, equivalence in $ {L_{\infty \kappa }}$
Article copyright: © Copyright 1974 American Mathematical Society

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