Reduction theorems for groups of matrices
type: Item
instance of: scholarly article
title: Reduction theorems for groups of matrices
zbMATH Open document ID: 1054.15011
author: J. Bernik
author: Mitja Mastnak
author: Robert Guralnick
published in: Linear Algebra and its Applications
publication date: 2004-08-06T00:00:00Z
review text: Let \(G\) be either a compact Lie group or a linear algebraic group over an algebraically closed field \(k\). The authors prove results of the following kind: \(G\) has a given property if and only if every finite subgroup has that property. Theorem 4 asserts that this is true for the properties of being solvable, nilpotent or abelian provided that in the latter two cases; the unipotent radical, \(G^0\), of \(G\) is trivial when \(\text{char}(k) = 0\). Simple examples show that the last proviso is necessary and that the theorem does not extend to non-compact Lie groups. The essential step in proving Theorem 4 for a linear algebraic group \(G\) is to show that, when \(G^0\) is a torus, \(G\) contains a locally finite, Zariski dense subgroup. Theorem 7 is a refinement of this result in the case where \(\text{char}(k)= p > 0\): here, the conclusion holds without restriction on \(G^0\).
reviewed by: G. E. Wall
Mathematics Subject Classification ID: 15A21
Mathematics Subject Classification ID: 15A30
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 22E20
zbMATH DE Number: 2081927
zbMATH Keywords: reduction theorems
zbMATH Keywords: compact Lie group
zbMATH Keywords: linear algebraic group
zbMATH Keywords: finite subgroups
zbMATH Keywords: solvable
zbMATH Keywords: nilpotent
zbMATH Keywords: abelian
MaRDI profile type: Publication
full work available at URL: j.laa.2003.11.020
OpenAlex ID: W2096814408
cites work: Q3999894
cites work: Structure of algebras
cites work: Q4118027
cites work: A finiteness lemma, brauer's theorem and other iIrreducibility results
cites work: Simultaneous triangularization
cites work: Free subgroups in linear groups
DOI: 10.1016/J.LAA.2003.11.020
description: scientific article; zbMATH DE number 2081927