Groups whose real irreducible characters have degrees coprime to \(p\).
type: Item
instance of: scholarly article
title: Groups whose real irreducible characters have degrees coprime to \(p\).
zbMATH Open document ID: 1259.20007
author: I. M. Isaacs
author: Gabriel Navarro
published in: Journal of Algebra
publication date: 2012-10-29T00:00:00Z
review text: This paper is concerned with variations of the well-known result of Itô and Michler that a finite group \(G\) has an Abelian normal Sylow \(p\)-subgroup if and only if \(p\) does not divide any irreducible (complex) character degree of \(G\). A recent result of Tiep (no reference is given in the paper) states that if every real-valued irreducible character of the finite group \(G\) has \(p'\)-degree for an odd prime \(p\), then \(O^{p'}(G)\) is solvable. In the paper under review, the authors study what else can be said about the groups satisfying the hypothesis of Tiep's result, and they prove a number of results on this. They show, for example, that if we put \(K=O^{2'}(G)\), then \(K\) has a normal Sylow \(p\)-subgroup \(Q\), and \(Q'\) is central in \(K\). Also, if \(G/K\) has a normal Sylow \(p\)-subgroup, then \(G\) has a normal Sylow \(p\)-subgroup, and if \(P\) is a Sylow \(p\)-subgroup of \(G\), then \(N_G(P)\) contains an element in each real conjugacy class of \(G\). Furthermore, a canonically defined bijection between the set of real-valued irreducible characters of \(G\) and the set of real-valued irreducible characters of \(p'\)-degree of \(N_G(P)\) is established.
reviewed by: Thomas Michael Keller
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20D20
zbMATH DE Number: 6099540
zbMATH Keywords: finite groups
zbMATH Keywords: Itô-Michler theorem
zbMATH Keywords: real characters
zbMATH Keywords: Sylow subgroups
zbMATH Keywords: character degrees
zbMATH Keywords: real-valued irreducible characters
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2012.02.007
OpenAlex ID: W2067123329
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DOI: 10.1016/J.JALGEBRA.2012.02.007
description: scientific article; zbMATH DE number 6099540