Pure subgroups of torsionfree groups
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 by Paul Hill and Charles Megibben PDF
 Trans. Amer. Math. Soc. 303 (1987), 765778 Request permission
Abstract:
In this paper, we show that certain new notions of purity stronger than the classical concept are relevant to the study of torsionfree 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.References

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Additional Information
 © Copyright 1987 American Mathematical Society
 Journal: Trans. Amer. Math. Soc. 303 (1987), 765778
 MSC: Primary 20K20; Secondary 20K27
 DOI: https://doi.org/10.1090/S00029947198709027979
 MathSciNet review: 902797