Parabolic Systems : the Gf ( 3 ) Case

Parabolic systems defined over GF(q) have been classified by Timmesfeld for q > 4 and by Stroth for q = 2 (see references). We deal with the case q = 3 . Parabolic systems have been classified by Niles, Timmesfeld, Stroth, and Heiss, if the field of definition is G F (2) or has at least four elements. [Ni, Timl, Tim2, Tim5, Tim7, Stl, St2, St3, He]. We treat the GF(3) case, where only partial results by Thiel exist so far [Th]. Our result says that strong parabolic systems in characteristic 3 have spherical diagram, and therefore essentially generate only finite groups of Lie type with the same diagram. This is the content of Theorem A. If we drop the assumption that the parabolic systems have to be strong, some infinite families of systems occur, whose diagrams are or complete bipartite graphs with only double or triple bonds, and the systems are classified. This is Theorem B. The results of this paper are used in the determination of locally finite classical Tits chamber systems with a transitive group of automorphisms having finite chamber stabilizers. This classification, in turn, could be used in the proof of the theorem of Kantor, Liebler, and Tits that determines all classical affine buildings of rank at least 3 having a discrete chamber-transitive group of automorphisms. The organization of the paper is as follows. The proof of Theorem A is given in §3, while the proof of Theorem B is contained in §4. Definitions, notation and some preliminaries are given in §1, while in §2 the relevant F .F-modules for some Lie-type groups defined over G F (3) axe determined. 1. Definitions, notation, preliminaries We are mainly concerned with characteristic 3, hence our flotation and the definitions reflect this fact. Let G be a finite group, we set G := O3'(G), and G = G/Oi(G). If S is a subgroup of G, by S g we denote the largest normal subgroup of G contained in S. If {Xx, i £ 1} is a system of subgroups of G, we set Xy for the group generated by X¡ and X¡. If X is a finite simple group of Lie type, PSL2(3) or 2G2(3), or a direct product of such groups (these Received by the editors August 3, 1989 and, in revised form, April 16, 1991. 1991 Mathematics Subject Classification. Primary 20D25, 51D20. © 1993 American Mathematical Society 0002-9947/93 $1.00+ $.25 per page