Butler groups of infinite rank. II
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- by Manfred Dugas, Paul Hill and K. M. Rangaswamy PDF
- Trans. Amer. Math. Soc. 320 (1990), 643-664 Request permission
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 $|A| \leqslant {\aleph _\omega }$. This implies that under CH all balanced subgroups of completely decomposable groups of cardinality $\leqslant {\aleph _\omega }$ are ${B_2}$-groups.References
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Additional Information
- © Copyright 1990 American Mathematical Society
- 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