Finite nonsolvable groups whose character graphs have no triangles.
type: Item
instance of: scholarly article
title: Finite nonsolvable groups whose character graphs have no triangles.
zbMATH Open document ID: 1207.20005
author: Yanjun Liu
author: Xueling Song
author: Tianze Li
published in: Journal of Algebra
publication date: 2010-04-21T00:00:00Z
review text: It is shown that the alternating group \(A_5\) is the only nonsolvable group whose character graph has no triangles. Here the character graph of a finite group \(G\) is defined as follows. The vertices of that graph are the nonlinear complex irreducible characters of \(G\), there is an edge between two vertices \(\chi\) and \(\psi\) if and only if \(\chi(1)\) and \(\psi(1)\) admit a common prime divisor. The authors mention that, due to \textit{O. Manz}, \textit{R. Staszewski} and \textit{W. Willems} [Proc. Am. Math. Soc. 103, No. 1, 31-37 (1988; Zbl 0645.20005)], the number of components of the character graph of any group is at most 3 and that it is at most 2 for any solvable group. Recently, due to \textit{Y.-T. Wu} and \textit{P. Zhang} [J. Algebra 308, No. 2, 536-544 (2007; Zbl 1118.20012)], a characterization of those finite solvable groups has been obtained whose character graphs have no triangles. Namely this happens precisely when the solvable group \(G\) is not isomorphic to \(S_4\) or if \(\#(\text{Irr}(G)\setminus\text{LIrr}(G))\leq 2\). Thus, by means of the paper under review and by the papers mentioned before, the classification of those finite groups whose character graphs have no triangles, is complete.
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20D05
zbMATH DE Number: 5696950
zbMATH Keywords: character graphs
zbMATH Keywords: degrees of irreducible characters
zbMATH Keywords: finite groups
zbMATH Keywords: irreducible complex characters
zbMATH Keywords: connected components
zbMATH Keywords: alternating group \(A_5\)
reviewed by: Robert W. van der Waall
describes a project that uses: CHEVIE
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2010.01.019
OpenAlex ID: W2019439690
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DOI: 10.1016/J.JALGEBRA.2010.01.019
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description: scientific article; zbMATH DE number 5696950