Butler groups of infinite rank
A torsion-free abelian group \(G\) is said to be a \(B_1\)-group if \(\text{Bext}^1(G,T) = 0\) for all torsion groups \(T\), where \(\text{Bext}^1(G,T)\) denotes the group of all balanced extensions of \(T\) by \(G\). Finite rank \(B_1\)-groups have been extensively studied under the name Butler groups. The \(B_2\)-groups form a subclass of \(B_1\)-groups possessing a smooth filtration of the so-called decent subgroups with rank-1 quotient groups and an important open question is whether every \(B_1\)-group is indeed a \(B_2\)-group. Utilizing the two key concepts of a relative balanced-projective resolution and of an \(\aleph_0\)-prebalanced subgroup, the author makes a striking progress in the study of \(B_1\)-groups. \(B_2\)-groups are characterized among \(B_1\)-groups by the property of possessing an \(\aleph_0\)-prebalanced chain. The last property for a torsion-free group \(G\) is elegantly connected to the kernel \(K\) in a balanced-projective resolution \(0 \to K \to C \to G \to 0\) of \(G\) being itself a \(B_2\)-group. The reviewer showed [Proc. Am. Math. Soc. 121, No. 2, 409-415 (1994; Zbl 0805.20045)] that, under the Continuum Hypothesis \((2^{\aleph_0} = \aleph_1)\), a torsion-free abelian group \(G\) is a \(B_2\)-group exactly when \(\text{Bext}^1(G,T)=\text{Bext}^2(G,T) = 0\) for all torsion groups \(T\). The author extends this by showing that, under the hypothesis, \(2^{\aleph_0} = \aleph_n\) for some integer \(n \geq 1\), a torsion-free \(G\) is a \(B_2\)-group if and only if \(\text{Bext}^1 (G,T) = \text{Bext}^2 (G,T) = \dots = \text{Bext}^n (G,T) = 0\). The powerful tools developed by the author enable him to completely settle the question as to which pure subgroups of a \(B_2\)-groups are again \(B_2\). There are several really interesting corollaries. This is a well-written paper. The author's use of the \(\aleph_0\)-prebalanced subgroups in the study of Butler groups may be termed a breakthrough and, naturally, it leads him prove some deep and interesting theorems.
relative balanced-projective resolutions
scientific article; zbMATH DE number 737478
scientific article