Hyperabelian skew-linear groups
type: Item
instance of: scholarly article
title: Hyperabelian skew-linear groups
zbMATH Open document ID: 0546.20045
DOI: 10.1016/0021-8693(84)90209-6
published in: Journal of Algebra
publication date: 1984-01-01T00:00:00Z
review text: For any group G let hph(G) be the least ordinal \(\gamma\) such that G is the union of \(\gamma\) normal subgroups containing each other and with locally nilpotent subsequent factors. Also hah(G) is defined similarly for abelian factors. Of course, hph(G)\(\leq hah(G).\) Theorem 1. Let \(p=0\) or a prime and let 1\(\leq\beta \leq\gamma \) be ordinals. Then there exists a division ring D of characteristic p that is locally finite-dimensional over its centre and a torsion-free hyperabelian subgroup G of \(D^* (=D-\{0\})\) such that \(hph(G)=\beta\) and \(hah(G)=\gamma\). - Let D be as theorem 1 and \(G\subset GL(n,D)\) with \(hph(G)=1\). Then it is known that \(hah(G)<\infty\). Let \(d(n,p)=\max\{hap(G)| G\subset GL(n,D),hph(G)=1\}.\) Theorem 2. \(d(n,p)=2+[\log_ 2(n)]\) if \(n=1\) or \(p=0\) and \(=1+[-\log_ 2(n)]\) if \(n>1\) and \(p>0\).
Mathematics Subject Classification ID: 20H25
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 16Kxx
Mathematics Subject Classification ID: 20E25
Mathematics Subject Classification ID: 20G35
Mathematics Subject Classification ID: 20E15
Mathematics Subject Classification ID: 16S50
zbMATH DE Number: 3869608
zbMATH Keywords: union of normal subgroups
zbMATH Keywords: division ring
zbMATH Keywords: locally finite-dimensional
zbMATH Keywords: torsion-free hyperabelian subgroup
author: B. A. F. Wehrfritz
MaRDI profile type: Publication
full work available at URL: 0021-8693(84)90209-6
OpenAlex ID: W2042125987
cites work: Varieties of groups
cites work: Observations on group rings
cites work: Q5676405
cites work: Nilpotence in Certain Linear Groups Over Division Rings
cites work: The upper central factors of certain skew-linear groups
cites work: On certain soluble skew-linear groups
cites work: Complete Reducibility in Skew Linear Groups
cites work: The structure of several classes of matrix groups over a division ring
description: scientific article; zbMATH DE number 3869608