The third axiom of countability for abelian groups

Hill, Paul

doi:10.1090/S0002-9939-1981-0612716-0

The third axiom of countability for abelian groups
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Proc. Amer. Math. Soc. 82 (1981), 347-350 Request permission

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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