A new characterization of Suzuki-Ree group
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.
scientific article; zbMATH DE number 1091424