On the character degree graph of solvable groups. I: Three primes
type: Item
instance of: scholarly article
title: On the character degree graph of solvable groups. I: Three primes
zbMATH Open document ID: 0930.20007
DOI: 10.1023/A:1004659919371
author: Péter P. Pálfy
published in: Periodica Mathematica Hungarica
publication date: 1999-10-31T00:00:00Z
review text: The character degree graph \(\Gamma(G)\) of a finite group is defined as follows: The vertices are those primes \(p\) that divide \(\chi(1)\) for some irreducible character \(\chi\) of \(G\). Two different vertices \(p,q\in\Gamma(G)\) are connected by an edge if there exists an irreducible character \(\chi\) with \(pq\mid\chi(1)\). Suppose that \(G\) is solvable. \textit{O. Manz, W. Willems} and \textit{T. R. Wolf} [J. Reine Angew. Math. 402, 181-198 (1989; Zbl 0678.20002)] have proved that whenever \(\pi\) is a subset of the vertices of \(\Gamma(G)\) with \(|\pi|=4\) then there exists a \(\chi\in\text{Irr}(G)\) which is divisible by at least 2 primes in \(\pi\). In the paper under review this result has been improved by showing the same conclusion but assuming only \(|\pi|=3\).
reviewed by: Wolfgang Willems
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 05C25
zbMATH DE Number: 1355146
zbMATH Keywords: character degrees
zbMATH Keywords: finite solvable groups
zbMATH Keywords: character degree graphs
Wikidata QID: Q106072091
MaRDI profile type: Publication
full work available at URL: a:1004659919371
OpenAlex ID: W1957823418
description: scientific article; zbMATH DE number 1355146