Autoequivalences of blocks and a conjecture of Zassenhaus
type: Item
instance of: scholarly article
title: Autoequivalences of blocks and a conjecture of Zassenhaus
zbMATH Open document ID: 0834.20005
DOI: 10.1016/0022-4049(94)00091-V
author: Frauke Bleher
author: Gerhard Hiss
author: Wolfgang Kimmerle
published in: Journal of Pure and Applied Algebra
publication date: 1996-01-30T00:00:00Z
review text: The authors deal with the Zassenhaus conjecture for certain finite simple groups. For a finite group \(G\) its integral group ring is denoted by \(\mathbb{Z} G\). The Zassenhaus conjecture claims that \(\mathbb{Z} G=\mathbb{Z} H\) as augmented algebras implies that \(a\cdot G\cdot a^{-1}=H\) for a unit \(a\in\mathbb{Q} G\). The authors verify this conjecture for \(\text{PSL}(2,p)\) with \(p\) a rational prime, for the five simple Mathieu groups, for \(A_9\) and for the sporadic Thompson group. For simple groups, the Zassenhaus conjecture is equivalent to the existence of a Zassenhaus factorization of augmented automorphisms of \(\mathbb{Z} G\): Given an augmented automorphism \(\alpha\) of \(\mathbb{Z} G\), then it has a factorization \(\alpha=\rho\cdot\gamma\), where \(\rho\in\text{Aut}(G)\) and \(\gamma\) is a central automorphism of \(\mathbb{Z} G\); i.e. it is conjugation with a unit in \(\mathbb{Q} G\) centralizing \(\mathbb{Z} G\). The proof is an interesting interplay between ordinary and modular representation theory: \(\alpha\) has a Zassenhaus factorization if and only if it acts like a group automorphism on the character table of \(G\). In order to verify this in the special examples modular representation theory is invoked: Let \(B\) be a block of \({\mathfrak k} G\) of cyclic defect (\(\mathfrak k\) must be large enough). If \(\alpha : B\to B\) is an automorphism fixing a module corresponding to an end of the Brauer-tree, then it fixes all finitely generated modules of \(B\). (Reviewer's remark: This result is definitely false if one replaces \(\mathfrak k\) by a large enough \(p\)-adic ring of integers.) This observation, together with an analysis of the character tables and decomposition matrices allows one to verify the Zassenhaus conjecture for the above mentioned groups.
reviewed by: Klaus W. Roggenkamp
Mathematics Subject Classification ID: 20C10
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 16U60
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 16S34
zbMATH DE Number: 810245
zbMATH Keywords: Zassenhaus conjecture
zbMATH Keywords: finite simple groups
zbMATH Keywords: integral group rings
zbMATH Keywords: augmented algebras
zbMATH Keywords: units
zbMATH Keywords: Mathieu groups
zbMATH Keywords: sporadic Thompson group
zbMATH Keywords: Zassenhaus factorizations
zbMATH Keywords: augmented automorphisms
zbMATH Keywords: character tables
zbMATH Keywords: blocks
zbMATH Keywords: cyclic defect groups
zbMATH Keywords: Brauer trees
zbMATH Keywords: finitely generated modules
zbMATH Keywords: decomposition matrices
Wikidata QID: Q122943304
describes a project that uses: GAP
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 810245