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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Pure subgroups of torsion-free groups

Authors: Paul Hill and Charles Megibben
Journal: Trans. Amer. Math. Soc. 303 (1987), 765-778
MSC: Primary 20K20; Secondary 20K27
MathSciNet review: 902797
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Abstract: In this paper, we show that certain new notions of purity stronger than the classical concept are relevant to the study of torsion-free abelian groups. In particular, implications of ${\ast }$-purity, a concept introduced in one of our recent papers, are investigated. We settle an open question (posed by Nongxa) by proving that the union of an ascending countable sequence of ${\ast }$-pure subgroups is completely decomposable provided the subgroups are. This result is false for ordinary purity. The principal result of the paper, however, deals with $\Sigma$-purity, a concept stronger than ${\ast }$-purity but weaker than the usual notion of strong purity. Our main theorem, which has a number of corollaries including the recent result of Nongxa that strongly pure subgroups of separable groups are again separable, states that a $\Sigma$-pure subgroup of a $k$-group is itself a $k$-group. Among other results is the negative resolution of the conjecture (valid in the countable case) that a strongly pure subgroup of a completely decomposable group is again completely decomposable.

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Article copyright: © Copyright 1987 American Mathematical Society