Square character degree graphs yield direct products.
type: Item
instance of: scholarly article
title: Square character degree graphs yield direct products.
zbMATH Open document ID: 1251.20011
author: Mark L. Lewis
author: Qingyun Meng
published in: Journal of Algebra
publication date: 2012-05-23T00:00:00Z
full work available at URL: https://arxiv.org/abs/1105.2799
reviewed by: Thomas Michael Keller
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 20D10
zbMATH DE Number: 6037573
zbMATH Keywords: character degree graphs
zbMATH Keywords: character degrees
zbMATH Keywords: finite solvable groups
zbMATH Keywords: squares
zbMATH Keywords: direct products
zbMATH Keywords: irreducible complex characters
MaRDI profile type: Publication
OpenAlex ID: W1969944202
arXiv ID: 1105.2799
cites work: Q4012093
cites work: Q3597669
cites work: Q4103728
cites work: Determining group structure from sets of irreducible character degrees
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cites work: Solvable groups whose degree graphs have two connected components
cites work: Bounding Fitting heights of character degree graphs
cites work: Q4273959
cites work: On the character degree graph of solvable groups. II. Discon- nected graphs
cites work: Q4428096
DOI: 10.1016/J.JALGEBRA.2011.09.016
review text: For any finite group \(G\), the character degree graph \(\Delta(G)\) is the graph whose vertices are the primes dividing some (irreducible complex) character degree of \(G\), and two such primes \(p,q\) are connected by an edge if \(pq\) divides some character degree. It is easy to find solvable groups whose character degree graph is a square. However, all known examples are direct products.NEWLINENEWLINE The main result of the paper under review is that indeed no other examples exist. More precisely, for a finite group \(H\) we denote by \(\rho(H)\) the set of all primes dividing some character degree of \(H\). If \(G\) is a finite solvable group for which \(\Delta(G)\) is a square, then there exist subgroups \(A\) and \(B\) of \(G\) such that \(G=A\times B\) and \(|\rho(A)|=|\rho(B)|=2\). The proof is, not unexpectedly, long and technical. The authors believe that the result remains true without the solvability hypothesis.
description: scientific article; zbMATH DE number 6037573