Bounded essential extensions of completely decomposable groups
type: Item
instance of: scholarly article
title: Bounded essential extensions of completely decomposable groups
zbMATH Open document ID: 0957.20034
published in: Journal of Algebra
publication date: 2000-12-14T00:00:00Z
review text: This paper concerns bcd groups, i.e. torsion-free Abelian groups of infinite rank which contain a completely decomposable subgroup of bounded index. These bcd groups are natural generalizations of the well-known acd groups, i.e. groups of the same description with infinite replaced by finite. One of the most significant concepts in the study of acd groups is that of regulating subgroup, which is defined in two equivalent ways: as a completely decomposable subgroup of minimal index, or as a direct sum of homogeneous completely decomposable subgroups obtained from Butler decompositions of the type subgroups. Now it turns out that in the infinite rank case, these definitions make sense but are no longer equivalent, and only the second one is useful. The authors show that all bcd groups have regulating subgroups in this second sense, but the quotient groups are not as controlled as in the acd case. They reveal the most important properties of regulating subgroups and their quotients, find numerous interesting examples and present several unsolved problems, of which the most important seems to be whether an infinite rank direct summand of a bcd group is a bcd group.
Mathematics Subject Classification ID: 20K20
Mathematics Subject Classification ID: 20K25
Mathematics Subject Classification ID: 20K27
zbMATH DE Number: 1496010
zbMATH Keywords: almost completely decomposable Abelian groups
zbMATH Keywords: co-Butler decompositions
zbMATH Keywords: Butler decompositions
zbMATH Keywords: torsion-free Abelian groups of infinite rank
zbMATH Keywords: completely decomposable subgroups of bounded index
zbMATH Keywords: bcd groups
zbMATH Keywords: acd groups
zbMATH Keywords: regulating subgroups
zbMATH Keywords: direct summands
author: Adolf Mader
author: Lutz Strüngmann
reviewed by: Phillip Schultz
MaRDI profile type: Publication
full work available at URL: jabr.2000.8306
OpenAlex ID: W2006786594
cites work: Q4346751
cites work: Almost Completely Decomposable Torsion Free Abelian Groups
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cites work: Improved model selection criterion
cites work: Q4315596
cites work: Q4270338
DOI: 10.1006/JABR.2000.8306
description: scientific article; zbMATH DE number 1496010