Characterizing normal Sylow \(p\)-subgroups by character degrees.
type: Item
instance of: scholarly article
title: Characterizing normal Sylow \(p\)-subgroups by character degrees.
zbMATH Open document ID: 1277.20010
author: Gunter Malle
author: Gabriel Navarro
published in: Journal of Algebra
publication date: 2013-05-13T00:00:00Z
review text: Let \(G\) be a finite group, \(p\) a prime number and \(P\) a Sylow \(p\)-subgroup of \(G\). The well-known Itô-Michler theorem states that \(P\) is Abelian and normal in \(G\) if and only if every irreducible character of \(G\) has degree not divisible by \(p\). In this paper the authors answer the question whether some canonical subset of the irreducible characters of \(G\) exactly captures the normality of \(P\) in the affirmative as follows: \(P\) is normal in \(G\) if and only if \(p\) does not divide the degree of any irreducible constituent of the permutation character \((1_P)^G\). They also show that this condition is equivalent to the property that \(\chi(x)\neq 0\) for all \(x\in P\) and all irreducible constituents \(\chi\) of the permutation character \((1_P)^G\). The proofs depend on CFSG.
reviewed by: Thomas Michael Keller
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 6162687
zbMATH Keywords: irreducible characters
zbMATH Keywords: finite groups
zbMATH Keywords: character degrees
zbMATH Keywords: Ito-Michler theorem
zbMATH Keywords: irreducible constituents
zbMATH Keywords: permutation character
zbMATH Keywords: normal Sylow subgroups
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2012.07.050
OpenAlex ID: W2070067422
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DOI: 10.1016/J.JALGEBRA.2012.07.050
description: scientific article; zbMATH DE number 6162687