Bounding graph diameters of solvable groups.
type: Item
instance of: scholarly article
title: Bounding graph diameters of solvable groups.
zbMATH Open document ID: 1077.20010
author: John K. McVey
published in: Journal of Algebra
publication date: 2005-02-22T00:00:00Z
review text: For a finite group \(G\) let \(\text{cd}(G)=\{\chi(1)\mid\chi\in\text{Irr}_\mathbb{C}(G)\}\). Associated to \(\text{cd}(G)\) we define two graphs \(\Gamma(G)\) and \(\Delta(G)\). The vertices of \(\Gamma(G)\) are \(\text{cd}(G)\setminus\{1\}\) and two vertices \(\chi(1)\) and \(\psi(1)\) are joined by an edge if \(\gcd(\chi(1),\psi(1))>1\). The graph \(\Delta(G)\) has \(\{p\mid p\) a prime and \(p\mid\chi(1)\) for some \(\chi\in\text{Irr}_\mathbb{C}(G)\}\) as the set of vertices and two primes \(p,q\) are adjacent if there is \(\chi(1)\in\text{cd}(G)\) with \(pq\mid\chi(1)\). Both graphs share some properties in common, for instance the number of connected components. Thus for many problems it does not matter which graph we choose for our investigations. In 1989 Manz, Wolf and the reviewer proved that for a solvable group \(G\) the diameter \(\text{diam}(\Delta(G))\) of a connected component is always bounded by 3 and recently (2002) Lewis could show that this bound is tight. From this, easy arguments lead to \(\text{diam}(\Gamma(G))\leq 4\). However, in the paper under review the author proves the better bound \(\text{diam}(\Gamma(G))\leq 3\), in parallel to the other graph \(\Delta(G)\). In the last section examples show that there are also interesting questions for which the parallelism between the two graphs fails.
reviewed by: Wolfgang Willems
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20D10
zbMATH DE Number: 2136727
zbMATH Keywords: graphs related to character degrees
zbMATH Keywords: diameters
zbMATH Keywords: solvable groups
zbMATH Keywords: irreducible character degrees
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2003.02.010
OpenAlex ID: W2010139370
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DOI: 10.1016/J.JALGEBRA.2003.02.010
description: scientific article; zbMATH DE number 2136727