Embeddings of \(\text{PGL}_2(31)\) and \(\text{SL}_2(32)\) in \(E_8(\mathbb{C})\). With appendices by Michael Larsen and J. -P. Serre
The authors extend work begun by the first author and \textit{A.~M.~Cohen} [Proc. Symp. Pure Math. 47, Pt. 2, 367-405 (1987; Zbl 0654.22005)] to classify the finite simple subgroups of \(E_8(\mathbb{C})\). They announce a program to classify embeddings of \(\text{PGL}_2(31)\), \(\text{SL}_2(32)\), \(\text{PSL}_2(31)\), \(\text{PSL}_2(41)\), \(\text{PSL}_2(49)\) and \(Sz(8)\) into \(E_8(\mathbb{C})\), and begin in the current paper with the first two cases. The authors show that \(E_8(\mathbb{C})\) has three conjugacy classes of \(\text{PGL}_2(31)\)-subgroups, which yield three nonconjugate embeddings of the latter group into the former. Two of these were previously obtained by \textit{J.-P.~Serre} [Invent. Math. 124, No. 1-3, 525-562 (1996; Zbl 0877.20033)], who in an appendix to the present paper points out that the authors' results imply that \(\text{PGL}_2(31)\) embeds into \(E_8(p)\) for every prime \(p\). The authors also show that \(E_8(\mathbb{C})\) has a single conjugacy class of \(\text{SL}_2(32)\)-subgroups, and Serre's appendix notes that this implies \(E_8(q)\) contains \(\text{SL}_2(32)\) if the prime power \(q\) is conjugate to \(\pm 1\) modulo 11. The authors use a computational approach, beginning with consideration of a finite group \(L\) and prime \(p\) that does not divide the order of \(L\). They show that the number of conjugacy classes of \(L\)-subgroups in a quasisimple algebraic group in characteristic 0 is bounded by the number of conjugacy classes of \(L\)-subgroups in a corresponding algebraic group in characteristic \(p\). (An appendix by M.~Larsen uses algebraic-geometric methods to show in fact that equality actually holds in general.) The authors use their preliminary results to classify embeddings of \(\text{PGL}_2(31)\) and then \(\text{PSL}_2(32)\) into \(E_8(\mathbb{C})\) by studying embeddings into algebraic groups of type \(E_8\) in specially chosen prime characteristics: respectively, 2791 and 4093. They systematically describe the associated computations representing the action of \(G=E_8(2791)\) on its 248-dimensional \(F_{2791}\)-module \(\mathcal E\), a Lie algebra of type \(E_8\). For example, in the first case they follow the procedure of \textit{R.~W.~Carter} [Simple groups of Lie type, Wiley, New York (1972; Zbl 0248.20015)] to obtain 16 generators of \(G\).
scientific article; zbMATH DE number 1425132
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