Sharp characters and prime graphs of finite groups
type: Item
instance of: scholarly article
title: Sharp characters and prime graphs of finite groups
zbMATH Open document ID: 0815.20007
author: N. Iiyori
published in: Journal of Algebra
publication date: 1995-06-30T00:00:00Z
review text: Let \(G\) be a finite group and \(\chi\) a virtual character of \(G\). Then \(| G|\) divides \(\prod_{x \in L^*} (\chi(1) - x)\), where \(L^* := \{\chi(g) \mid g\in G\setminus \{1\}\}\). We say that \(\chi\) is sharp (of type \(L^*\)) if \(| G| = \prod_{x \in L^*} (\chi(1) - x)\). In this paper relations between the structure of \(G\) and its sharp characters are studied, also in connection with graph theory. The main purpose of the paper is to prove the following Theorem 1. Let \(G\) be a finite group. Then equivalent are: (a) \(2 \leq \text{com}(G)\); (b) \(G\) has a 2-connected sharp ordinary character of rank 2; (c) \(G\) has a 2-connected character of rank 2; (d) the augmentation ideal of \(G\) decomposes as a module. There are more results proven, but we cannot mention them here. Finally we mention for the convenience of the reader: Definition. The prime graph \(\Gamma(G)\) of the finite group \(G\) is the following graph: \(V(\Gamma(G)) = \pi(G)\) and \(xy \in E(\Gamma(G))\) if and only if there exists \(g \in G\) such that \(x \cdot y\) divides \(| g|\). We denote by \(\text{com}(G)\) the number of connected components of \(\Gamma(G)\).
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20C05
zbMATH DE Number: 501075
zbMATH Keywords: finite group
zbMATH Keywords: virtual character
zbMATH Keywords: sharp characters
zbMATH Keywords: sharp ordinary characters
zbMATH Keywords: augmentation ideal
zbMATH Keywords: prime graph
reviewed by: Robert W. van der Waall
MaRDI profile type: Publication
full work available at URL: jabr.1994.1001
OpenAlex ID: W1975684335
DOI: 10.1006/JABR.1994.1001
description: scientific article; zbMATH DE number 501075