A direct product coming from a particular set of character degrees.
type: Item
instance of: scholarly article
zbMATH Open document ID: 1251.20008
author: Kamal Aziziheris
publication date: 2012-05-07T00:00:00Z
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 6031520
zbMATH Keywords: representations of finite groups
zbMATH Keywords: character degrees of finite groups
zbMATH Keywords: direct products of groups
zbMATH Keywords: Brauer characters
zbMATH Keywords: Frobenius groups
zbMATH Keywords: degree graphs
zbMATH Keywords: finite solvable groups
zbMATH Keywords: sets of character degrees
zbMATH Keywords: irreducible complex characters
MaRDI profile type: Publication
full work available at URL: jgt.2011.083
OpenAlex ID: W2067898285
cites work: Determining group structure from the sets of character degrees.
cites work: Determining group structure from sets of irreducible character degrees
cites work: Solvable groups whose degree graphs have two connected components
cites work: Prime Divisibility Among Degrees of Solvable Groups
cites work: Groups having three complex irreducible character degrees
DOI: 10.1515/JGT.2011.083
title: A direct product coming from a particular set of character degrees.
published in: Journal of Group Theory
review text: It is proved that if \(G\) is a solvable group with \(\text{cd}(G)=\{1,m,p,q,mp,mq\}\), where \(p\) and \(q\) are distinct primes and \(m>1\) is an integer not divisible by \(p\) or \(q\), then \(G=A\times B\), where \(\text{cd}(A)=\{1,p,q\}\) and \(\text{cd}(B)=\{1,m\}\). This generalizes [\textit{M. L. Lewis}, J. Algebra 206, No. 1, 235-260 (1998; Zbl 0915.20003), Theorem A]. Here \(\text{cd}(G)\) stands for the set of the degrees of all the irreducible complex characters of \(G\). Lewis in his Theorem A did considers the case \(m\) is a prime, different from \(p\) and \(q\). The proof of the main theorem of the author established here uses a lot of Brauer character theory, Frobenius action on vector spaces, degree graphs and Frobenius groups. As such, the proof of the main theorem is certainly not an easy thing to overcome; it is very nice.
reviewed by: Robert W. van der Waall
description: scientific article; zbMATH DE number 6031520