Infinite rank Butler groups
Authors:
Manfred Dugas and K. M. Rangaswamy
Journal:
Trans. Amer. Math. Soc. 305 (1988), 129142
MSC:
Primary 20K20; Secondary 20K35, 20K40
DOI:
https://doi.org/10.1090/S0002994719880920150X
MathSciNet review:
920150
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Abstract  References  Similar Articles  Additional Information
Abstract: A torsionfree 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 socalled ${B_2}$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
Keywords:
Torsionfree abelian groups,
Butler groups,
pure subgroups
Article copyright:
© Copyright 1988
American Mathematical Society