1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

CHAPTER 12 Larger groups over F 2 in .Cj(G, T) In this chapter we consider the cases remaining in the Fundamental Setup (3.2.1) ...

788 12. LARGER GROUPS OVER F2 IN .Cj(G, T) an F 2 L-module. Let T 1 := Nr(V 1 ), and for v EV, let Mv := CM(v), Gv := Ca(v), and ...

12.1. A PRELIMINARY CASE: ELIMINATING Ln(2) ON n Ell n* 789 so that Tv E Syh(H), B = 02 (H), and Q E Syh(QB). As B 1. M = !M(LT) ...

790 i2. LARGER GROUPS OVER F2 IN .Cj(G, T) A is faithful on each 1/i, and for u E Vi - Cv 1 (A), [u, Al = [Vi, Al is of rank 2. ...

i2.1. A PRELIMINARY CASE: ELIMINATING Ln(2) ON n EEl n* 79i parabolic of K* determined by the end nodes of the Dynkin diagram. T ...

792 12. LARGER GROUPS OVER F2 IN .Cj(G, T) So assume instead that Vi Qv. Then by the Baer-Suzuki Theorem, there is g E Gv such t ...

12.1. A PRELIMINARY CASE: ELIMINATING L 0 (2) ON n EB n* 793 ~^9 with fixed axis B n ~^9 for i = 1 and 2. Thus as Vi is dual to ...

794 12. LARGER GROUPS OVER F2 IN .C'f (G, T) totally singular subspace of Cv(X), dim(Cv(X)) 2 2dim(U). Thus dim(V/U) 2 dim(V/Cv( ...

12.2. GROUPS OVER F2, AND THE CASE V A TI-SET IN G 795 THEOREM 12.2.2. Assume Hypothesis 12.2.1. Then one of the following holds ...

796 12. LARGER GROUPS OVER F2 IN .Cj(G, T) one of the conclusions of Theorem 12.2.2 holds-namely (a), (b ), the subcase of (d) w ...

12.2. GROUPS OVER F2, AND THE CASE V A TI-SET IN G 797 LEMMA 12.2.6. V is a TI-set in M, so if 1 # U ::=:; V and H ::=:; Na(U), ...

798 12. LARGER GROUPS OVER Fz IN .Cj(G, T) M = Na(X 0 ) since Na(Xo) = !M(XoT) by 1.3.7, so H::::; YT::::; Na(Xo) ::::; M, contr ...

12.2. GROUPS OVER F2, AND THE CASE V A TI-SET IN G 799 may apply A.3.15. However, the list of possibilities for L/0 2 (L) from A ...

soo 12. LARGER GROUPS OVER F2 IN .Cj(G, T) (1) Mv = Gv ~Gu= Mu. Next if u"' = v for some x EM, then gx E Gv:::::; M, so g E Mx-^ ...

12.2. GROUPS OVER F2, AND THE CASE VA TI-SET ING 801 Suppose that An02(H) = 1. Then A~ A, so m(A) = m(V) 23by12.2.2.3. But if H ...

802 12. LARGER GROUPS OVER F2 IN .Cj(G, T) the noncentral chief factors for K on Vz are A 5 -modules.^1 Therefore case (ii) of 1 ...

12.2. GROUPS OVER F2, AND THE CASE V A TI-SET IN G 803 with U = [W,J] = [W, h]. Further [W, V] = [U, V] = V n U = V n Wis of ran ...

804 12. LARGER GROUPS OVER F2 IN .Cj(G, T) Furthemore m(W) :::; 3m(U n V) = 6, so m(W) = 6, since this is the minimal dimension ...

12.2. GROUPS OVER F2, AND THE CASE V A TI-SET IN G 805 PROOF. Let R := Cr(U). By a Frattini Argument, R = UNR(P), and then R = U ...

806 12. LARGER GROUPS OVER F2 IN .Cj(G, T) I :S: Na(R) :S: N 0 (Cr(L)) :S: M = !Jvl(LT), contrary to I 1:. M. Thus Cr(L) = 1, so ...