Butler groups of infinite rank
type: Item
instance of: scholarly article
title: Butler groups of infinite rank
zbMATH Open document ID: 0842.20045
DOI: 10.1016/0022-4049(95)90015-2
published in: Journal of Pure and Applied Algebra
publication date: 1996-07-23T00:00:00Z
review text: 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 an…
Mathematics Subject Classification ID: 20K20
Mathematics Subject Classification ID: 20K35
Mathematics Subject Classification ID: 20K40
Mathematics Subject Classification ID: 20K27
zbMATH DE Number: 737478
zbMATH Keywords: torsion-free Abelian groups
zbMATH Keywords: balanced extensions
zbMATH Keywords: \(B_ 1\)-groups
zbMATH Keywords: Butler groups
zbMATH Keywords: \(B_ 2\)-groups
zbMATH Keywords: smooth filtrations
zbMATH Keywords: decent subgroups
zbMATH Keywords: relative balanced-projective resolutions
zbMATH Keywords: \(\aleph_ 0\)-prebalanced subgroups
zbMATH Keywords: \(\aleph_ 0\)-prebalanced chains
zbMATH Keywords: pure subgroups
author: László Fuchs
reviewed by: Kulumani M. Rangaswamy
MaRDI profile type: Publication
full work available at URL: 0022-4049(95)90015-2
OpenAlex ID: W2075379846
cites work: Butler groups of infinite rank and axiom 3
cites work: On the Dimension of Modules and Algebras (III): Global Dimension
cites work: Q3686899
cites work: On abelian groups by which balanced extensions of a rational group split
cites work: Subgroups of butler groups
cites work: Q3661801
cites work: A Class of Torsion-Free Abelian Groups of Finite Rank
cites work: Butler Groups of Infinite Rank II
cites work: Infinite Rank Butler Groups
cites work: Separable pure subgroups of completely decomposable torsion-free abelian groups
cites work: The Functor Bext under the Negation of CH
cites work: Note on extensions of Abelian groups by primary groups
cites work: Butler groups of arbitrary cardinality
cites work: Q4025946
cites work: Corank one subgroups of completely decomposable abelian groups
cites work: Butler groups that are unions of subgroups with countable typesets
cites work: Q3918299
cites work: The Third Axiom of Countability for Abelian Groups
cites work: Pure Subgroups of Torsion-Free Groups
cites work: A Homological Characterization of Abelian B 2 -Groups
description: scientific article; zbMATH DE number 737478