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

DOI:
https://doi.org/10.1090/S0002-9947-1987-0902797-9

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 -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 -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 -purity, a concept stronger than -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 -pure subgroup of a -group is itself a -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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DOI:
https://doi.org/10.1090/S0002-9947-1987-0902797-9

Article copyright:
© Copyright 1987
American Mathematical Society