Finite groups with graphs containing no triangles
type: Item
instance of: scholarly article
title: Finite groups with graphs containing no triangles
zbMATH Open document ID: 1024.20021
DOI: 10.1016/S0021-8693(03)00142-X
author: Ming Fang
author: Pu Zhang
published in: Journal of Algebra
publication date: 2003-07-30T00:00:00Z
review text: For a given finite group \(G\) define a graph \(\Gamma(G)\) by taking the non-central conjugacy classes of \(G\) as vertices and connecting any two vertices \(A\) and \(B\) whenever \(|A|\) and \(|B|\) have a common prime divisor. By studying the properties of \(\Gamma(G)\) it is expected to gain some useful information about \(G\) and vice versa. The purpose of the paper is to prove the following Theorem. Let \(G\) be a non-Abelian finite group. Then \(\Gamma(G)\) is a graph without triangles if and only if \(G\) is isomorphic to one of the following solvable groups: (1) the symmetric group \(S_3\); (2) a dihedral group of order \(10\) or \(12\); (3) the alternating group \(A_4\); (4) the group \(T_{12}:=\langle a,b\mid a^6=1,\;b^2=a^3,\;ba=a^{-1}b\rangle\) of order \(12\); (5) the group \(T_{21}:=\langle a,b\mid a^3=b^7=1,\;ba=ab^2\rangle\) of order \(21\). Furthermore, the property that \(\Gamma(G)\) has no triangles is equivalent to the one that \(\Gamma(G)\) is a disjoint union of two connected trees.
reviewed by: Wolfgang Lempken
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 20E45
zbMATH DE Number: 1956753
zbMATH Keywords: conjugacy classes
zbMATH Keywords: graphs
zbMATH Keywords: Frobenius groups
zbMATH Keywords: finite groups
MaRDI profile type: Publication
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full work available at URL: s0021-8693(03)00142-x
OpenAlex ID: W1984314029
description: scientific article; zbMATH DE number 1956753