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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Pure subgroups of torsion-free groups
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by Paul Hill and Charles Megibben PDF
Trans. Amer. Math. Soc. 303 (1987), 765-778 Request permission


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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Additional Information
  • © Copyright 1987 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 303 (1987), 765-778
  • MSC: Primary 20K20; Secondary 20K27
  • DOI:
  • MathSciNet review: 902797