transactions of the american mathematical society Volume 320, Number 2, August 1990 BUTLER GROUPS OF INFINITE RANK II MANFRED DUGAS, PAUL HILL, AND K. M. RANGASWAMY Abstract. A torsion-free abelian group G is called a Butler group if Bext(G, 7") = 0 for any torsion group T . We show that every Butler group G of cardi- nality N, is a ¿?2"8rouP; 'e-' C7 is a union of a smooth ascending chain of pure subgroups Ga where Ga+X = Ga + Ba , Ba a Butler group of finite rank. Assuming the validity of the continuum hypothesis (CH), we show that every Butler group of cardinality not exceeding N^ isa ß2-group. Moreover, we are able to prove that the derived functor Bext (A , T) = 0 for any torsion group T and any torsion-free A with \A\ < Kw . This implies that under CH all balanced subgroups of completely decomposable groups of cardinality < Hm are 52"ßrouPs- 1. Introduction All groups in this paper are (torsion-free) abelian groups. Undefined nota- tions are standard as in [Fu]. A torsion-free abelian group is called a Butler group if Bext(B, T) = 0 for all torsion groups T. Here Bext is the subfunctor of Ext of all balanced-exact extensions. It is known [BS] that this definition coincides with the familiar one if B has finite rank, i.e., a pure subgroup of a completely decomposable group. The main result in this paper is that under the continuum hypothesis (CH) each Butler group B (of rank < Nw) is a B2-group [A]; i.e., B is the union of a smooth ascending chain of pure subgroups Ba , a < k an ordinal and B ,. = B +L for all a < X where L is a Butler group a+l a a a or of finite rank. This result partially answers questions raised in [BSS], [A], and elsewhere. We will, in fact, prove the following: Theorem [CH], Let G be a torsion-firee group of cardinality < Kw. (a) The following are equivalent: (1) Bext(t7, T) = 0 for all torsion groups T. (2) Bext(C7, T) = 0 for all 1,-cyclic torsion groups T. (3) G is a B2-group. (ß) The derived function Bext (G, T) = 0 for any torsion group T. (We have to use CH only if \G\ > N2.) Received by the editors October 13, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 20K20. The research of the first author was partially supported by NSF Grant DMS 8701074. The research of the second author was partially supported by NSF Grant DMS 8521770. ©1990 American Mathematical Society 0002-9947/90 $1.00+ $.25 per page 643 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use