Inequalities for some blocks of finite groups.
type: Item
instance of: scholarly article
title: Inequalities for some blocks of finite groups.
zbMATH Open document ID: 1109.20011
author: Gunter Malle
author: Gabriel Navarro
published in: Archiv der Mathematik
publication date: 2006-12-06T00:00:00Z
review text: Let \(G\) be a finite group and \(B\) be a \(p\)-block of \(G\) with defect group \(D\). Let \(k(B)=|\text{Irr}(B)|\), \(l(B)=|\text{IBr}(B)|\), \(k_0(B)\) be the number of height zero characters in \(B\), and \(k(D)\) be the number of irreducible characters of \(D\). It is proved that the inequalities \[ k(B)/k_0(B)\leq k(D')\quad\text{ and }\quad k(B)/l(B)\leq k(D)\tag{1} \] hold under each of the following hypotheses: (i) \(D\) is normal in \(G\), (ii) \(G\) is a symmetric group, or (iii) \(G\) is an exceptional group of Lie type in its defining characteristic. The authors wonder whether similar inequalities might be true for general blocks. This might be of interest as, for example, even the weaker inequality \(k(B)/k_0(B)\leq|D'|\) together with the Alperin-McKay Conjecture would imply Brauer's \(k(B)\)-conjecture. However, the authors express some skepticism that (1) could be true for all \(p\)-solvable groups.
reviewed by: Thomas Michael Keller
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20C33
Mathematics Subject Classification ID: 20C30
zbMATH DE Number: 5077611
zbMATH Keywords: finite groups
zbMATH Keywords: blocks
zbMATH Keywords: numbers of characters
zbMATH Keywords: defect groups
zbMATH Keywords: Alperin-McKay conjecture
zbMATH Keywords: Brauer \(k(B)\)-conjecture
zbMATH Keywords: \(p\)-solvable groups
MaRDI profile type: Publication
full work available at URL: s00013-006-1769-8
OpenAlex ID: W1967099394
DOI: 10.1007/S00013-006-1769-8
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description: scientific article; zbMATH DE number 5077611