Finite Groups of Seitz Type

We show that a useful condition of Seitz on finite groups of Lie type over fields of order q > 4 is often satisfied when q is 2 or 3. We also observe that various consequences of the Seitz condition, established by Seitz and Cline, Parshall, and Scott when q > 4, also hold when q is 3 or 4. In this paper we explore a certain property shared by most finite groups of Lie type first exploited by Seitz in [Se1]. In order to state our results precisely we need to be a bit careful with notation and terminology. So let p be a prime, and define a Lie simple group of Lie type of characteristic p to be a central factor group Ω of a group Ḡσ, where Ḡ is a simply connected simple algebraic group over the algebraic closure F̄p of the field of order p and σ is a Steinberg endomorphism of Ḡ with finite fixed points. Thus Ω is a central factor group of a simply connected classical group SLn(q), Spn(q), Spinn(q), SUn(q), or of a simply connected exceptional group G2(q), F4(q), Ẽ6(q), Ẽ7(q), E8(q), D4(q), Ẽ6(q), B2(q), G2(q), or F4(q), for some power q of p. In particular, Ω is quasisimple unless Ω is an image of SL2(2), SL2(3), SU3(2), Sp4(2), G2(2), B2(2), G2(3), or F4(2). Moreover, we say that Ω is defined over Fq. A nearly simple group of Lie type of characteristic p is a finite group G possessing a normal subgroup Ω, which is a Lie simple group of Lie type in characteristic p, and such that CG(Ω) = Z(G) is a p ′-group. In addition we adopt the following notation: Let Φ be a root system for Ω, Φ a positive system with simple system π, and (Ω, BΦ, NΦ, S) the Tits system determined by Φ and Φ. Set HΦ = BΦ ∩ NΦ, the Cartan subgroup of Ω determined by Φ and Φ, U = Rad(BΦ), and for α ∈ Φ let Uα be the root subgroup of α. Set B = NG(U); we call B the Borel subgroup of G, and we call the overgroups of B in G the standard parabolic subgroups of G. The parabolic subgroups of G are the conjugates of the standard parabolics. Recall that U ∈ Sylp(Ω), so G = ΩB by a Frattini argument. Set H = B ∩B0 and N = NΦH, where w0 is the long word in the Weyl group W = NΦ/HΦ ∼= N/H. Assume q is a power of the prime p, G is a nearly simple group of Lie type, Ω is quasisimple, defined over Fq, and is not G2(q), and AutG(Ω) is a group of inner-diagonal automorphisms of Ω. We define such a group G to be of Seitz type if Received by the editors February 23, 2012 and, in revised form, March 7, 2012 and March 9, 2012. 2010 Mathematics Subject Classification. Primary 20D05, 20E42. This work was partially supported by NSF grants DMS-0504852 and DMS-0969009. c ©2013 American Mathematical Society Reverts to public domain 28 years from publication