A group theoretic characterization of linear groups
type: Item
instance of: scholarly article
title: A group theoretic characterization of linear groups
zbMATH Open document ID: 0647.20045
DOI: 10.1016/0021-8693(88)90190-1
author: Alexander Lubotzky
published in: Journal of Algebra
publication date: 1988-01-01T00:00:00Z
review text: The author gives a group-theoretical characterization of finitely generated linear groups over a field of characteristic zero. Let \(\Gamma\) be an abstract group. A \(p\)-congruence structure (with bound \(c\)) for \(\Gamma\) is a descending series of finite index normal subgroups of \(\Gamma =N_ 0\supseteq N_ 1\supseteq N_ 2\supseteq ...\) such that (i) \(\cap_{i=0}^{\infty} N_ i = \{1\}\); (ii) \(N_ 1/N_ i\) is a \(p\)-group for every \(i>1\); (iii) \(N_ i/N_ j\) can be generated by \(c\) elements for every \(j > i > 1\). Theorem. Let \(\Gamma\) be a finitely generated group. Then the following conditions are equivalent: (A) \(\Gamma\) is isomorphic to a matrix group over \(\mathbb{C}\). (B) \(\Gamma\) has a \(p\)-congruence structure for some prime \(p\).
Mathematics Subject Classification ID: 20G20
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20H20
zbMATH DE Number: 4055902
zbMATH Keywords: finitely generated linear groups
zbMATH Keywords: \(p\)-congruence structure
zbMATH Keywords: descending series
zbMATH Keywords: finite index normal subgroups
MaRDI profile type: Publication
full work available at URL: 0021-8693(88)90190-1
OpenAlex ID: W2080182681
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description: scientific article; zbMATH DE number 4055902