Sylow normalizers and Brauer character degrees
type: Item
instance of: scholarly article
title: Sylow normalizers and Brauer character degrees
zbMATH Open document ID: 0968.20008
author: Gabriel Navarro
author: Antonio Beltrán
published in: Journal of Algebra
publication date: 2001-06-10T00:00:00Z
review text: Let \(p\) and \(q\) be primes and let \(G\) be a finite \(\{p,q\}\)-solvable group. Let \(P\) be a Sylow \(p\)-subgroup of \(G\) and \(Q\) a Sylow \(q\)-subgroup of \(G\). Let \(N_G(P)\) and \(N_G(Q)\) denote the normalizers in \(G\) of these subgroups. The main result of the paper under review is the following. Each irreducible \(p\)-modular Brauer character of \(G\) that has \(q'\)-degree also has \(p'\)-degree if and only if \(N_G(Q)\) is contained in some conjugate of \(N_G(P)\). The authors also give examples to show that their main theorem is not true without some solvability hypothesis. As a corollary, they prove that it is possible to decide from the character table of \(G\) whether or not \(N_G(Q)\) is contained in a conjugate of \(N_G(P)\). The main tool in the proof is Clifford theory, combined with the delicate techniques of analysis of characters of (generalized) solvable groups developed by I. M. Isaacs and G. Navarro.
reviewed by: Rod Gow
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20D10
zbMATH DE Number: 1502167
zbMATH Keywords: character degrees
zbMATH Keywords: \(\pi\)-solvable groups
zbMATH Keywords: Sylow normalizers
zbMATH Keywords: \(p\)-Brauer characters
zbMATH Keywords: character tables
MaRDI profile type: Publication
full work available at URL: jabr.1999.8275
OpenAlex ID: W2087221174
cites work: Characters of \(\pi\)-separable groups
cites work: Q4871631
cites work: Q4839675
cites work: Two groups with isomorphic group algebras
cites work: Actions and invariant character degrees
cites work: Q4387775
cites work: Character degrees and local subgroups of 𝜋-separable groups
DOI: 10.1006/JABR.1999.8275
description: scientific article; zbMATH DE number 1502167