Simple groups which are 2-fold OD-characterizable.
type: Item
instance of: scholarly article
title: Simple groups which are 2-fold OD-characterizable.
zbMATH Open document ID: 1241.20020
author: Juan-Miguel Gracia
published in: Bulletin of the Malaysian Mathematical Sciences Society. Second Series
publication date: 2012-01-26T00:00:00Z
full work available at URL: vol35 1 7.html
review text: The well-known notion of the prime graph \(\Gamma(G)\) of a finite group \(G\) has many important applications to the study of arithmetical properties of finite groups. In particular, results on the prime graph of finite simple groups are used in the investigations of the problem of the recognition of finite almost simple groups by their orders and some properties of their prime graphs. Let \(G\) be a finite group and \(\pi(|G|)=\{p_1,\dots,p_k\}\) with \(p_1<p_2<\cdots<p_k\). For \(p\in\pi(|G|)\), let \(\deg(p)\) be the degree (valency) of the vertex \(p\) in the graph \(\Gamma(G)\). According to \textit{A. R. Moghaddamfar, A. R. Zokayi} and \textit{M. R Darafsheh} [Algebra Colloq. 12, No. 3, 431-442 (2005; Zbl 1072.20015)], we define \(D(G)=(\deg(p_1),\dots,\deg(p_k))\), which is called the degree pattern of \(G\). The group \(G\) is called \(k\)-fold OD-characterizable if exist exactly \(k\) non-isomorphic finite groups with the same order and the same pattern as \(G\). Usually, a 1-fold OD-characterizable group is simply called an OD-characterizable group. In the present paper, the authors first prove that each simple group from the pair \(\{B_n(q),C_n(q)\}\) is 2-fold OD-characterizable in the following cases: \(n=2^m>2\), \(q\) is odd and \(|\pi((q^n+1)/2)|=1\); \(n\) is an odd prime, \(q=3\) and \(|\pi((3^n-1)/2)|=1\); \(n=3\) and \(q=5\). They also prove that groups \(B_n(q)\) and \(C_n(q)\) are OD-characterizable in the following cases: \(n=2\) and \(|\pi((q^2+1)/(2,q-1)|=1\); \(n=2^m\geq 2\), \(q\) is even and \(|\pi((q^n+1))|=1\).
reviewed by: Anatoli Kondrat'ev
Mathematics Subject Classification ID: 20D06
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 05C25
zbMATH DE Number: 6000386
zbMATH Keywords: finite groups
zbMATH Keywords: simple groups
zbMATH Keywords: symplectic groups
zbMATH Keywords: orthogonal groups
zbMATH Keywords: spectra
zbMATH Keywords: prime graphs
zbMATH Keywords: degree patterns
zbMATH Keywords: OD-characterizability
zbMATH Keywords: degrees of vertices
zbMATH Keywords: order components
zbMATH Keywords: sets of element orders
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 6000386