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

 

The third axiom of countability for abelian groups


Author: Paul Hill
Journal: Proc. Amer. Math. Soc. 82 (1981), 347-350
MSC: Primary 20K10; Secondary 20K15, 20K20
MathSciNet review: 612716
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Abstract: Three different definitions of the third axiom of countability for abelian $ p$-groups are shown to be equivalent. The main interest in this stems from the fact that the third axiom of countability characterizes one of the most important classes of abelian groups. Moreover, the equivalence of two of these definitions validates the proof of Theorem 67 in P. Griffith's Infinite abelian group theory.

As a further application of our method of proof, we show that every torsionfree abelian group satisfies the third axiom of countability with respect to purity.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1981-0612716-0
PII: S 0002-9939(1981)0612716-0
Keywords: Third axiom of countability, totally projective group, ascending chain of nice subgroups, Ulm invariant, torsionfree group, pure subgroup
Article copyright: © Copyright 1981 American Mathematical Society