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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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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.
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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