On nilpotent blocks of finite groups
type: Item
instance of: scholarly article
title: On nilpotent blocks of finite groups
zbMATH Open document ID: 0796.20008
author: Atumi Watanabe
published in: Journal of Algebra
publication date: 1994-10-11T00:00:00Z
review text: Let \(G\) be a finite group, and let \(B\) be a block of \(G\) with defect group \(D\). If \(s(B)\) denotes the number of conjugacy classes of \(B\)- Brauer elements then, in general, \(s(B) \leq k(D)\) where \(k(D)\) denotes the number of conjugacy classes of \(D\). The author shows that \(B\) is nilpotent if and only if \(s(B) = k(D)\). Suppose that Alperin's weight conjecture holds for all sections of \(G\). Then it is shown that \(B\) is nilpotent if and only if, for every \(B\)-Brauer pair \((Q,b_ Q)\), \(b_ Q\) contains exactly one irreducible Brauer character.
reviewed by: Burkhard Külshammer
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20C05
zbMATH DE Number: 501082
zbMATH Keywords: nilpotent blocks
zbMATH Keywords: Brauer pair
zbMATH Keywords: Brauer elements
zbMATH Keywords: finite group
zbMATH Keywords: defect group
zbMATH Keywords: number of conjugacy classes
zbMATH Keywords: Alperin's weight conjecture
zbMATH Keywords: irreducible Brauer character
MaRDI profile type: Publication
full work available at URL: jabr.1994.1008
OpenAlex ID: W2167773019
DOI: 10.1006/JABR.1994.1008
description: scientific article; zbMATH DE number 501082