Rational matrix groups of a special type
type: Item
instance of: scholarly article
title: Rational matrix groups of a special type
zbMATH Open document ID: 0575.20005
DOI: 10.1016/0024-3795(85)90255-1
author: Peter Schmid
published in: Linear Algebra and its Applications
publication date: 1985-01-01T00:00:00Z
review text: Let the finite group G have a faithful irreducible complex character \(\chi\) such that \(\chi\) (1) is relatively prime to \(| G| /\chi (1)\). Let \(F=Q(\chi)\) and let \(n=[F:Q]\chi (1)\). Then G embeds in \(GL_ n(Q)\), with centralizer isomorphic to \(GL_ 1(F)\). The author shows that if \(\chi\) (1) and [F:Q] are not both even, then the normalizer of G in \(GL_ n(Q)\) splits over the centralizer. The proof uses cohomology and Clifford theory. The theorem was motivated by the special case of Chevalley groups of adjoint type, with \(\chi\) the Steinberg character.
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20J05
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20G20
zbMATH DE Number: 3918630
zbMATH Keywords: irreducible complex character
zbMATH Keywords: normalizer
zbMATH Keywords: centralizer
zbMATH Keywords: Clifford theory
zbMATH Keywords: Chevalley groups
zbMATH Keywords: Steinberg character
reviewed by: William H. Gustafson
MaRDI profile type: Publication
full work available at URL: 0024-3795(85)90255-1
OpenAlex ID: W2033737140
cites work: The Schur subgroup. II
cites work: Some Properties of Characters of Finite Groups
cites work: Extending irreducible representations of normal subgroups
cites work: Q5619224
description: scientific article; zbMATH DE number 3918630