Finite groups of Seitz type. (Q2862175)

Let \(p\) be a prime, \(\overline G\) be a simply connected simple algebraic group over the algebraic closure of the field of order \(p\), \(\sigma\) be a Steinberg endomorphism of \(\overline G\) with finite group \(\overline G_\sigma\) of fixed points. Then a central factor group of \(\overline G\) is called a Lie simple group of Lie type of characteristic \(p\). Let \(G\) be a finite group possessing a normal subgroup \(\Omega\), which is a Lie simple group of Lie type defined over the field of order \(q\), where \(q\) is a power of the prime \(p\), and such that \(\Omega\) is quasisimple, \(\Omega\) is not \(^2G_2(q)\), \(C_G(\Omega)=Z(G)\) is a \(p'\)-group, and \(\Aut_G(\Omega)\) is a group inner-diagonal automorphisms of \(\Omega\).NEWLINENEWLINE Let \(\Phi\) be a root system for \(\Omega\), \(\Phi^+\) a positive system for \(\Phi\) and \((B_\Phi,N_\Phi)\) the natural BN-pair determined by \(\Phi\) and \(\Phi^+\), \(U\in\text{Syl}_p(B_\Phi)\) and \(U_\alpha\) the root subgroup of \(\Omega\) for \(\alpha\in\Phi\). Set \(B=N_G(U)\) and \(H=B\cap B^{w_0}\), where \(w_0\) is the longest word in the Weyl group of the BN-pair. Then \(G=\Omega B\) by a Frattini argument.NEWLINENEWLINE The author defines such a group \(G\) to be of Seitz type if (Se1) for each \(\alpha\in\Phi^+\), \(H\) is irreducible on \(U_\alpha/\Phi(U_\alpha)\) and \(Z(U_\alpha)\), and (Se2) for all distinct \(\alpha,\beta\in\Phi^+\), \(C_H(U_\alpha/\Phi(U_\alpha))\neq C_H(U_\beta/\Phi(U_\beta))\neq C_H(Z(U_\alpha))\neq C_H(Z(U_\beta))\). \textit{G. M. Seitz} [in Lemma 3 from J. Algebra 28, 508-517 (1974; Zbl 0338.20053)] proved essentially that if \(q>4\) then \(G\) is of Seitz type. When \(q=2\), \(G\) is almost never of Seitz type.NEWLINENEWLINE In this paper, the author proves that if \(G\) is not of Seitz type and \(q\in\{3,4\}\) then \(G/Z(G)\) is isomorphic to one of the following groups: \(L_3(4)\), \(G_2(4)\), \(L_4(3)\), \(U_4(3)\), \(U_4(3)\) extended by \(\mathbb Z_2\), \(\text{PSp}_n(3)\) with \(n\geq 4\), \(P\Omega_m^\varepsilon(3)\) with \(m\geq 7\), \(F_4(3)\), or \(G_2(3)\).NEWLINENEWLINE The usefulness of a result of similar form was noted by the reviewer in his survey [Russ. Math. Surv. 41, No. 1, 65-118 (1986); translation from Usp. Mat. Nauk 41, No. 1(247), 57-96 (1986; Zbl 0602.20041)]. The Seitz conditions have a number of useful consequences, some of which were established by \textit{G. Seitz} [in loc. cit.; J. Algebra 61, 16-27 (1979; Zbl 0426.20036) and Pac. J. Math. 106, 153-244 (1983; Zbl 0522.20031)], and \textit{E. Cline, B. Parshall} and \textit{L. Scott} [in J. Algebra 34, 521-523 (1975; Zbl 0324.20051)]. The author remarks that when \(q>4\) and \(G\) is not \(L_2(9)\), the fact that \(G\) is of Seitz type, together with an argument of \textit{H. N. Ward} [J. Algebra 10, 377-382 (1968; Zbl 0167.02305)], can be used to show that \(p\)-part of the Schur multiplier of a simple group \(G\) of Lie type over the field of order \(q\) is trivial. The author uses also the fact that most groups of Lie type are of Seitz type in his paper [J. Algebra 382, 71-99 (2013; Zbl 1293.20016)] and then in the program aimed at showing that certain finite lattices are not intervals in the subgroup lattices of finite groups.