Infinite rank Butler groups

Abstract

A torsion-free abelian group G G is said to be a Butler group if Bext ⁡ ( G , T ) \operatorname {Bext} (G,\,T) for all torsion groups T 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 G satisfies the T.E.P. over a pure subgroup H H if and only if H H is decent in G G in the sense of Albrecht and Hill. A subclass of the Butler groups are the so-called B 2 {B_2} -groups. An important question left open by Arnold, Bican, Salce, and others is whether every Butler group is a B 2 {B_2} -group. We show under ( V = L ) (V = L) that this is indeed the case for Butler groups of rank ℵ 1 {\aleph _1} . On the other hand it is shown that, under ZFC, it is undecidable whether a group B B for which Bext ⁡ ( B , T ) = 0 \operatorname {Bext} (B,\,T) = 0 for all countable torsion groups T T is indeed a B 2 {B_2} -group.