Finite groups of \(G_2(3)\)-type
The group \(G_2(3)\)~is the Chevalley group of type \(G_2\) over the field of order 3. A finite group \(G\) is said to be of \(G_2(3)\)-type if \(G\) has subgroups \(H\) and \(M\) such that (G1) \(H\) has normal subgroups \(H_1\) and \(H_2\) with \(H_1\cong H_2\cong\text{SL}_2(3)\), \(|H:H_1H_2|=2\), \(\mathbb{Z}_2\cong H_1\cap H_2\), and \(H=C_G(H_1\cap H_2)\); and (G2) \(H_1\cap H_2\leq V\triangleleft M\) with \(C_M(V)=V\cong E_8\) and \(M/V\cong L_3(2)\). The main theorem of the present paper is: If \(G\) is of \(G_2(3)\)-type then \(G\cong G_2(3)\). In the proof of the classification of the finite simple groups, the group \(G_2(3)\) arises as a quasithin group of characteristic 2. This class of groups is treated in an unpublished work of the author and S. Smith, where \(G_2(3)\) is identified using the main theorem. The existing characterizations of \(G_2(3)\) were obtained by \textit{J. G. Thompson} [Pac. J. Math. 33, 451-536 (1970; Zbl 0256.20024)], \textit{Z. Janko} [J. Algebra 12, 360-371 (1969; Zbl 0177.04001)], \textit{P. Fong} and \textit{W. J. Wong} [Nagoya Math. J. 36, 143-184 (1969; Zbl 0188.06402), ibid. 39, 39-79 (1970; Zbl 0188.06501)]. The purpose of the author here is to obtain a much shorter and simpler treatment for purposes of the classification, using modern methods which are more conceptual, avoid character theory, and minimize detailed computations. In particular, the method of amalgams (see the book of \textit{A. Delgado, D. Goldschmidt} and \textit{B. Stellmacher} [``Groups and graphs: new results and methods'', Birkhäuser, Basel (1985; Zbl 0566.20013)]) and an approach of \textit{H. Bender} [in Ill. J. Math. 18, 223-228 (1974; Zbl 0279.20015)] are used.
scientific article; zbMATH DE number 1868060
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