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Abstract. We show that the negation of the continuum hypothesis implies that the derived functor Bext2 is not zero and there exist balanced subgroups of completely decomposable groups of rank ^i that are not Butler groups. 1980 Mathematics Subject Classification (1985 Revision): 20K20. § 0. Introduction All groups in this paper are abelian and our notations are Standard s in [Fu I/II]. A torsion-free group B of arbitrary rank is called a Butler group if Bext1 (B, ) = 0 for any torsion group T. Here Bext1^, T) denotes the subgroup of Ext (B, T) of all balanced exact extensions. We refer to [Ar] for an excellent survey on Butler groups of infinite rank and to [AH], [DHR] for more recent results. P. Hill introduced the notion of a separable subgroup A of a (torsion-free) group G. (This is not to be confused with the notion of a group G being "separable" in the sense of R. Baer). The subgroup A of G is separable if for any coset 4- A A there is a countable subset C of A such that for all a e A there is a c e C with | -h a \
Forum Mathematicum – de Gruyter
Published: Jan 1, 1991
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