Character degree graphs with no complete vertices.
type: Item
instance of: scholarly article
title: Character degree graphs with no complete vertices.
zbMATH Open document ID: 1256.20006
author: Claudio Paolo Morresi Zuccari
published in: Journal of Algebra
publication date: 2012-08-23T00:00:00Z
reviewed by: Thomas Michael Keller
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 6070505
zbMATH Keywords: finite solvable groups
zbMATH Keywords: irreducible characters
zbMATH Keywords: character degrees
zbMATH Keywords: representations
zbMATH Keywords: character degree graphs
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2011.11.029
OpenAlex ID: W1977687942
cites work: Q4103728
cites work: Solvable groups whose degree graphs have two connected components
cites work: Bounding Fitting heights of character degree graphs
cites work: A solvable group whose character degree graph has diameter \(3\)
cites work: Q4273959
cites work: On the character degree graph of solvable groups. I: Three primes
cites work: On the character degree graph of solvable groups. II. Discon- nected graphs
cites work: Finite Solvable Groups Whose Character Degree Graphs Are Not Complete
DOI: 10.1016/J.JALGEBRA.2011.11.029
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review text: Let \(G\) be a finite group. The character degree graph \(\Gamma(G)\) of \(G\) is the graph with vertex set \(\rho(G)=\{p\mid p\text{ a prime, }p\mid\chi(1)\text{ for some }\chi\in\text{Irr}(G)\}\) and edges between two elements \(p\) and \(q\) of \(\rho(G)\) if and only if \(pq\) divides some irreducible character degree of \(G\).NEWLINENEWLINE The purpose of the article under review is to establish whether certain graphs can be character degree graphs of solvable groups and to obtain a characterization of the groups that admit them. The main result is the following. Let \(G\) be solvable with Abelian Fitting group \(F(G)\). Suppose that the degree in \(\Gamma(G)\) of every \(\upsilon\in\rho(G)\) is at most \(|\rho(G)-2|\). Then there is an integer \(n\) such that \(F(G)=M_1\times\cdots\times M_n\times Z\), where \(M_1,\dots,M_n\) are minimal normal subgroups of \(G\) and \(Z=Z(G)\); moreover \(G=D_1\times\cdots\times D_n\), where \(M_1\leq D_1,\dots,M_n\leq D_n\), \((|D_i/M_i|,|D_j/M_j|)=1\) for all \(i\neq j\), and \(\Gamma(D_1),\dots,\Gamma(D_n)\) are disconnected graphs.NEWLINENEWLINE The author draws some consequences of this result. For example, if \(G\) is a solvable group with Abelian Fitting subgroup and connected character degree graph, then the diameter of \(\Gamma(G)\) is at most 2. (The general best possible bound for the diameter in the case of solvable groups is known to be 3.)
description: scientific article; zbMATH DE number 6070505