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Transactions of the American Mathematical Society

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

 

 

Butler groups of infinite rank. II


Authors: Manfred Dugas, Paul Hill and K. M. Rangaswamy
Journal: Trans. Amer. Math. Soc. 320 (1990), 643-664
MSC: Primary 20K20; Secondary 20K40
DOI: https://doi.org/10.1090/S0002-9947-1990-0963246-8
MathSciNet review: 963246
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Abstract: A torsion-free abelian group $ G$ is called a Butler group if $ \operatorname{Bext} (G,T) = 0.$ for any torsion group $ T$. We show that every Butler group $ G$ of cardinality $ {\aleph _1}$ is a $ {B_2}$-group; i.e., $ G$ is a union of a smooth ascending chain of pure subgroups $ {G_\alpha }$ where $ {G_{\alpha + 1}} = {G_\alpha } + {B_\alpha },{B_\alpha }$ a Butler group of finite rank. Assuming the validity of the continuum hypothesis (CH), we show that every Butler group of cardinality not exceeding $ {\aleph _\omega }$ is a $ {B_2}$-group. Moreover, we are able to prove that the derived functor $ {\operatorname{Bext} ^2}(A,T) = 0$ for any torsion group $ T$ and any torsion-free $ A$ with $ \vert A\vert \leqslant {\aleph _\omega }$. This implies that under CH all balanced subgroups of completely decomposable groups of cardinality $ \leqslant {\aleph _\omega }$ are $ {B_2}$-groups.


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DOI: https://doi.org/10.1090/S0002-9947-1990-0963246-8
Article copyright: © Copyright 1990 American Mathematical Society