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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality 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 1.43.

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Infinite rank Butler groups
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by Manfred Dugas and K. M. Rangaswamy PDF
Trans. Amer. Math. Soc. 305 (1988), 129-142 Request permission

Abstract:

A torsion-free abelian group $G$ is said to be a Butler group if $\operatorname {Bext} (G, T)$ for all torsion groups $T$. It is shown that Butler groups of finite rank satisfy what we call the torsion extension property (T.E.P.). A crucial result is that a countable Butler group $G$ satisfies the T.E.P. over a pure subgroup $H$ if and only if $H$ is decent in $G$ in the sense of Albrecht and Hill. A subclass of the Butler groups are the so-called ${B_2}$-groups. An important question left open by Arnold, Bican, Salce, and others is whether every Butler group is a ${B_2}$-group. We show under $(V = L)$ that this is indeed the case for Butler groups of rank ${\aleph _1}$. On the other hand it is shown that, under ZFC, it is undecidable whether a group $B$ for which $\operatorname {Bext} (B, T) = 0$ for all countable torsion groups $T$ is indeed a ${B_2}$-group.
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Additional Information
  • © Copyright 1988 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 305 (1988), 129-142
  • MSC: Primary 20K20; Secondary 20K35, 20K40
  • DOI: https://doi.org/10.1090/S0002-9947-1988-0920150-X
  • MathSciNet review: 920150