Primes dividing the degrees of the real characters.
type: Item
instance of: scholarly article
title: Primes dividing the degrees of the real characters.
zbMATH Open document ID: 1149.20006
author: Silvio Dolfi
author: Gabriel Navarro
author: Pham Huu Tiep
published in: Mathematische Zeitschrift
publication date: 2008-06-30T00:00:00Z
review text: Let \(G\) be a finite group and \(P\in\text{Syl}(G)\). The main result of the paper proves that if all real-valued irreducible characters of \(G\) have odd degree, then \(P\) is normal in \(G\). This is a version of the Ito-Michler Theorem for real-valued characters and the prime 2. Its proof depends on CFSG, and the authors remark that it does not remain true for odd primes, but defer the study of this case to a later paper. The second main result of the paper studies a graph whose vertices are the primes dividing the degrees of the real-valued irreducible character degrees, and two such primes are joined by an edge if their product divides the degree of some real-valued irreducible character degree. They show that this graph has at most three connected components, and if \(G\) is solvable, it has at most two connected components. The paper also contains some results on similar questions for the real conjugacy class sizes of a group; the proofs here are quite a bit easier.
reviewed by: Thomas Michael Keller
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20E45
zbMATH DE Number: 5294687
zbMATH Keywords: real-valued characters
zbMATH Keywords: Ito-Michler theorem
zbMATH Keywords: character degree graphs
zbMATH Keywords: conjugacy classes
describes a project that uses: CHEVIE
describes a project that uses: GAP
MaRDI profile type: Publication
full work available at URL: s00209-007-0247-8
OpenAlex ID: W2145787156
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DOI: 10.1007/S00209-007-0247-8
description: scientific article; zbMATH DE number 5294687