Fitting heights and the character degree graph
type: Item
instance of: scholarly article
title: Fitting heights and the character degree graph
zbMATH Open document ID: 0986.20007
author: Mark L. Lewis
published in: Archiv der Mathematik
publication date: 2002-05-27T00:00:00Z
review text: Let \(G\) be a finite solvable group and \(\rho(G)=\bigcup_{\chi\in\text{Irr}(G)}\pi(\chi(1))\), where \(\pi(n)\) is the set of prime divisors of \(n\in\mathbb{N}\); let \(\text{cd}(G)=\{\chi(1)\mid\chi\in\text{Irr}(G)\}\). We can attach a graph \(\Gamma(G)\) to \(\text{cd}(G)\) called the degree graph of \(G\). The vertex set of this graph is \(\rho(G)\), and we place an edge between \(p\) and \(q\), if \(pq\) divides some degree \(a\in\text{cd}(G)\). A degree graph \(\Gamma\) has bounded Fitting height if there is a bound on the Fitting heights of the solvable groups whose degree graph is \(\Gamma\). Main theorem: Let \(\Gamma\) be a degree graph with \(n\) vertices. Then \(\Gamma\) has bounded Fitting height if and only if \(\Gamma\) has at most one vertex of degree \(n-1\). Theorems 3 and 4. Let \(G\) be solvable group with degree graph \(\Gamma=\Gamma(G)\), \(\Gamma\) has \(n\) vertices. (a) If \(\Gamma\) has at most one vertex of degree \(n-1\), then the Fitting height \(h(G)\leq 4(n-1)+2\). (b) Assume that \(\Gamma\) has distinct vertices \(u\) and \(v\) both having degree \(n-1\). Then \(\Gamma\) has unbounded Fitting height.
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 05C25
zbMATH DE Number: 1556584
zbMATH Keywords: finite solvable groups
zbMATH Keywords: degree graphs
zbMATH Keywords: Fitting heights
reviewed by: Yakov Berkovich
MaRDI profile type: Publication
full work available at URL: s000130050512
OpenAlex ID: W2036929636
DOI: 10.1007/S000130050512
description: scientific article; zbMATH DE number 1556584