Finite groups with three rational conjugacy classes
type: Item
instance of: scholarly article
title: Finite groups with three rational conjugacy classes
zbMATH Open document ID: 1390.20008
author: Dan Rossi
published in: Archiv der Mathematik
publication date: 2018-01-26T00:00:00Z
full work available at URL: http://hdl.handle.net/10150/627837
review text: \textit{G. Navarro} and \textit{P. H. Tiep} [Trans. Am. Math. Soc. 360, No. 5, 2443--2465 (2008; Zbl 1137.20009)] proved that for a finite group \(G\), \(|\mathrm{Irr}_{\mathbb{Q}}(G)|=1\) if and only if \(|\mathrm{Cl}_{\mathbb{Q}}(G)|=1\) and \(|\mathrm{Irr}_{\mathbb{Q}}(G)|=2\) if and only if \(|\mathrm{Cl}_{\mathbb{Q}}(G)|=2,\) where \(\mathrm{Irr}_{\mathbb{Q}}(G)\) denotes the set of irreducible \(\mathbb{Q}\)-characters of group \(G\) and \(\mathrm{Cl}_{Q}(G)\) denotes the set of \(\mathbb{Q}\)-classes of \(G.\) They conjunctured that for a finite group \(G,\) \(|\mathrm{Irr}_{\mathbb{Q}}(G)|=3\) if and only if \(|\mathrm{Cl}_{\mathbb{Q}}(G)|=3.\) In this paper, the author mentions that this is the best possible generalisation of the theorem of Navarro and Tiep [loc. cit.] and he proves one direction of this conjucture, i.e. for a finite group \(G\) holds \(|\mathrm{Irr}_{\mathbb{Q}}(G)|=3\) if \(|\mathrm{Cl}_{\mathbb{Q}}(G)|=3\). The proof uses the classification of finite simple groups.
reviewed by: Dilpreet Kaur
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20E45
zbMATH DE Number: 6830200
zbMATH Keywords: rational irreducible character
zbMATH Keywords: rational conjugacy class
MaRDI profile type: Publication
OpenAlex ID: W2772524803
cites work: Primes dividing the degrees of the real characters.
cites work: Q5634146
cites work: Q3934684
cites work: Group elements and fields of character values
cites work: Solvable groups having only three rational classes of 2-elements.
cites work: Real characters and degrees.
cites work: Rational irreducible characters and rational conjugacy classes in finite groups
cites work: Degrees of rational characters of finite groups.
cites work: 2-length and rational characters of odd degree.
DOI: 10.1007/S00013-017-1127-Z
description: scientific article; zbMATH DE number 6830200