Finite groups of the Lie type of small rank
type: Item
instance of: scholarly article
title: Finite groups of the Lie type of small rank
zbMATH Open document ID: 0930.20012
author: Frauke Bleher
published in: Pacific Journal of Mathematics
publication date: 1999-10-21T00:00:00Z
review text: Let \(G\) be a finite group. The Zassenhaus conjecture for integral group rings predicts that given two subgroups \(X\) and \(Y\) in the group of units of augmentation \(1\) of \(\mathbb{Z} G\), if the order of \(X\) and of \(Y\) equals the order of \(G\), then there is a unit \(a\) in \(\mathbb{Q} G\) so that \(aXa^{-1}=Y\). In case \(G\) has the property that any subgroup \(X\) as above is abstractly isomorphic to \(G\), then the conjecture is equivalent to the statement that any automorphism of \(\mathbb{Z} G\) preserving augmentation can be modified by a group automorphism of \(G\) so that the composite is the identity on the centre of \(\mathbb{Z} G\). Finite simple groups are such groups where one knows that the abstract isomorphism exists. False for solvable groups, the Zassenhaus conjecture remains open for finite simple groups. In the paper under review, the author proves the above Zassenhaus conjecture for all finite simple groups \(G\) of Lie type of rank \(1\) and of rank \(2\) which are not isomorphic to \(\text{PSU}_4(q^2)\) or \(\text{PSU}_5(q^2)\). Moreover, the author proves the conjecture for all finite simple groups with Abelian \(2\)-Sylow subgroups. The method of proof is to look at the action of an automorphism preserving the augmentation on the \(p\)-modular character table, on the ordinary character table and on the Brauer tree. Then, the proof is a case by case analysis of the different Dynkin diagrams and Brauer trees.
Mathematics Subject Classification ID: 20C33
Mathematics Subject Classification ID: 16U60
Mathematics Subject Classification ID: 16W20
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 20C20
zbMATH DE Number: 1344294
zbMATH Keywords: finite simple groups of Lie type
zbMATH Keywords: integral group rings
zbMATH Keywords: groups of units
zbMATH Keywords: Zassenhaus conjecture
zbMATH Keywords: character tables
zbMATH Keywords: Brauer trees
reviewed by: Alexander Zimmermann
MaRDI profile type: Publication
full work available at URL: pjm.1999.187.215
OpenAlex ID: W2083503668
Wikidata QID: Q115230681
DOI: 10.2140/PJM.1999.187.215
description: scientific article; zbMATH DE number 1344294