Infinite rank Butler groups

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.