Characterization of \(^3D_4(q)\)
Let \(G\) be a finite group and \(\pi_i\) (\(1\leq i\leq t\)) be 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 author called these integers order components of \(G\) and gave a characterization of the following finite simple groups by their order components: sporadic simple groups [Algebra Colloq. 3, No. 1, 49-58 (1996; Zbl 0845.20011)], \(G_2(3^n)\) [J. Southwest China Normal Univ. (1996)], Suzuki-Ree simple groups [Sci. China, Ser. A 40, No. 8, 807-812 (1997; Zbl 0890.20015)], \(E_8(q)\) [J. Southwest China Normal Univ. 21, No. 3, 215-217 (1997)], \(\text{PSL}_2(q)\) [Southeast Asian Bull. Math. 22, No. 3, 257-263 (1998; Zbl 0936.20010)]. In this paper, the author proves that if a finite group \(G\) has the same order components as \(^3D_3(q)\) then \(G\cong {^3D_3(q)}\). 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)]). This result implies that, for the group \(^3D_3(q)\), Thompson's conjecture and Shi's conjecture (see the problems 12.37 and 12.39 in [``Kourovka Notebook'', 12th edition, Novosibirsk (1992; Zbl 0831.20003)], respectively) are true.
scientific article; zbMATH DE number 1807984