A new characterization of Suzuki-Ree group
type: Item
instance of: scholarly article
title: A new characterization of Suzuki-Ree group
zbMATH Open document ID: 0890.20015
DOI: 10.1007/BF02878919
published in: Science in China. Series A
publication date: 1998-04-23T00:00:00Z
review text: Let \(G\) be a finite group and \(\pi_i\) (\(1\leq i\leq t\)) are all prime graph components of \(G\). Then \(|G|\) can be expressed as a product of coprime positive integers \(m_1,\dots,m_t\), where \(\pi(m_i)=\pi_i\) (\(1\leq i\leq t\)). The set \(\{m_1,\dots,m_t\}\) is denoted by \(OC(G)\). The author proves the following theorem: if \(G\) is a finite group, and \(M\) is one of the groups \(^2B_2(q)\), \(^2G_2(q)\), \(^2F_4(q)\) (Suzuki-Ree groups) with \(OC(G)=OC(M)\), then \(G\cong M\). The proof uses the classification of prime graph components of (known) finite simple groups (see \textit{J. S. Williams} [J. Algebra 69, 487-513 (1981; Zbl 0471.20013)]\ and the reviewer [Mat. Sb. 180, No. 6, 787-797 (1989; Zbl 0691.20013)]). As corollaries of this theorem, the two characterizations of Suzuki-Ree groups proved by \textit{W. J. Shi} and \textit{J. X. Bi} [Sci. China, Ser. A 34, No. 1, 14-19 (1991; Zbl 0736.20011)]\ and by the author [J. Algebra 185, No. 1, 184-193 (1996; Zbl 0861.20018)]\ are obtained.
reviewed by: Anatoli Kondrat'ev
Mathematics Subject Classification ID: 20D06
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 20D08
Mathematics Subject Classification ID: 20E32
Mathematics Subject Classification ID: 20D20
zbMATH DE Number: 1091424
zbMATH Keywords: finite groups
zbMATH Keywords: Suzuki-Ree groups
zbMATH Keywords: orders of conjugacy classes
zbMATH Keywords: orders of elements
zbMATH Keywords: prime graph components
zbMATH Keywords: finite simple groups
author: Guiyun Chen
MaRDI profile type: Publication
cites work: A characterization of Suzuki-Ree groups
cites work: On Thompson's conjecture
cites work: Prime graph components of finite groups
cites work: PRIME GRAPH COMPONENTS OF FINITE SIMPLE GROUPS
cites work: Q4869726
cites work: Q3684278
description: scientific article; zbMATH DE number 1091424