A remark on Brauer's height zero conjecture
type: Item
instance of: scholarly article
title: A remark on Brauer's height zero conjecture
zbMATH Open document ID: 0848.20005
author: Masafumi Murai
published in: Journal of Mathematics of Kyoto University
publication date: 1996-10-28T00:00:00Z
review text: Let \(G\) be a finite group, \(p\) a prime and \(B\) be a \(p\)-block with defect group \(D\). Brauer's `height 0-conjecture' states: (HZ) Every irreducible character in \(B\) is of height 0 if and only if \(D\) is abelian. Let (AHZ) denote the `if'-part and (HZA) the `only if'-part of this conjecture. The assertion (AHZ) has been reduced to the case of quasi-simple groups by \textit{T. R. Berger} and \textit{R. Knörr} [Nagoya Math. J. 109, 109-116 (1988; Zbl 0637.20006)], whereas (HZA) has been proved for \(p\)-solvable groups by \textit{D. Gluck} and \textit{T. R. Wolf} [Trans. Am. Math. Soc. 282, 137-152 (1984; Zbl 0543.20007)]. Let \(k_0(B)\) denote the number of irreducible characters in \(B\) of height zero and \(\widetilde {B}\) be the Brauer correspondent of \(B\) in \(N_G(D)\), then the Alperin-McKay conjecture asserts: (AM) \(k_0(B)=k_0(\widetilde{B})\). In the present paper the author proves the following Theorem: Assume that (AM) is true for the principal blocks of all finite groups and that (HZA) is true for the principal blocks of all simple groups. Then (HZA) is true for the principal blocks of all finite groups.
reviewed by: Peter Fleischmann
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 871811
zbMATH Keywords: Brauer's height 0-conjecture
zbMATH Keywords: quasi-simple groups
zbMATH Keywords: number of irreducible characters
zbMATH Keywords: Brauer correspondence
zbMATH Keywords: Alperin-McKay conjecture
zbMATH Keywords: principal blocks
zbMATH Keywords: finite groups
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250518650
OpenAlex ID: W1542001752
Wikidata QID: Q122957923
DOI: 10.1215/KJM/1250518650
description: scientific article; zbMATH DE number 871811