TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 305, Number 1, January 1988 INFINITE RANKBUTLER GROUPS MANFRED DUGAS AND K. M. RANGASWAMY Abstract. A torsion-free abelian group G is said to be a Butler group if Bext(C, T) = 0 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 B2-groups. An important question left open by Arnold, Bican, Salce, and others is whether every Butler group is a ^-group. We show under ( V = L) that this is indeed the case for Butler groups of rank Nt. On the other hand it is shown that, under ZFC, it is undecidable whether a group B for which Bext( B, T) = 0 for all countable torsion groups T is indeed a B2-group. 1. Introduction. The class of completely decomposable torsion-free abelian groups is one of the few classes of abelian groups that can be completely determined by isomorphism invariants, as was shown by R. Baer [5]. In 1965 M. C. R. Butler [10] initiated the study of the pure subgroups of finite rank completely decomposable groups and in recent years this class of groups has been studied in greater detail by D. Arnold [2, 4], D. Arnold and C. Vinsonhaler [3], L. Bican [6, 7], L. Bican and L. Salce [8], and others. These finite rank torsion-free abelian groups have been called Butler groups and are shown to have several interesting characterizing properties including the following striking result by L. Bican [6]: A finite rank torsion-free abelian group G is a Butler group if and only if Bext(G, T) = 0 for all torsion groups T. Here Bext denotes the subfunctor of the functor Ext defined by the proper class of balanced exact sequences [13]. In their attempt to generalize the properties of (finite rank) Butler groups to infinite rank groups, Bican and Salce [8], Bican, Salce, and Stepan [9], and Arnold [4] were led to the following three classes of abelian groups: 3SX The class of torsion-free abelian groups G for which Bext(C7,T) = 0 for all torsion groups T Received by the editors January 5, 1987. Presented at The 93rd Annual Meeting of the American Mathematical Society at San Antonio, Texas, January 21, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 20K20; Secondary 20K35, 20K40. Key words and phrases. Torsion-free abelian groups, Butler groups, pure subgroups. Supported by a research grant from the College of Engineering and Applied Science, University of Colorado, Colorado Springs. ©1988 American Mathematical Society 0002-9947/88 $1.00 + $.25 per page 129 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use