The largest character degree and the Sylow subgroups of finite groups.
type: Item
instance of: scholarly article
title: The largest character degree and the Sylow subgroups of finite groups.
zbMATH Open document ID: 1062.20008
author: Guohua Qian
author: Shi Wujie
published in: Journal of Algebra
publication date: 2004-09-23T00:00:00Z
review text: Let \(P\) be a Sylow \(p\)-subgroup of a finite group \(G\) and let \(b(G)\) denote the largest degree of an irreducible complex character of \(G\). In the paper under review the authors show: For any finite group \(G\) the following bounds hold: a) \(|P/O_p(G)|<b(G)^2\). b) \(|P/O_p(G)|\leq b(G)\) if \(P\) is Abelian. The proof splits into two parts. Part one, which deals with the case that all minimal normal subgroups of \(G\) are solvable, is mainly based on a paper of \textit{I. M. Isaacs} [Proc. Am. Math. Soc. 127, No. 1, 45-50 (1999; Zbl 0916.20011)]. The other part, in which \(G\) has a non-solvable minimal normal subgroup, uses the classification of finite non-Abelian simple groups \(H\) and a careful analysis of the largest degree of \(H\) in relation to the order of the \(p\)-part of \(|\Aut(H)|\).
reviewed by: Wolfgang Willems
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 2102473
zbMATH Keywords: degrees of irreducible characters
zbMATH Keywords: Sylow subgroups
zbMATH Keywords: character degree bounds
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2003.02.009
OpenAlex ID: W1966542212
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DOI: 10.1016/J.JALGEBRA.2003.02.009
description: scientific article; zbMATH DE number 2102473