1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

14.7. FINISHING L 3 (2) WITH (Val) ABELIAN 1067 (2) QQH = Rl. Thus Q* = Ri = 02(Mi) is the unipotent radical of M{. (3) If H* ~ ...

1068 14. L 3 (2) IN THE FSU, AND L 2 (2) WHEN .Cf(G, T) IS EMPTY PROOF. Suppose first that U~ is not an FF*-offender on UH. As m ...

14.7. FINISHING Ls(2) WITH (vG1) ABELIAN 1069 Pick Fo E :F. As Vz and Fo are distinct hyperplanes of Fo V 1 ~ Es, Vz n Fo = Vf f ...

1070 14. L 3 (2) IN THE FSU, AND L 2 (2) WHEN .Cr(G, T) IS EMPTY LEMMA 14.7.65. VUo = Cu(R1) and V = [Cu(R1),L1]. In Notation 14 ...

14.7. FINISHING L 3 (2) WITH (VGl) ABELIAN 1071 Next suppose g E G with V{ ::::; E. As K centralizes V 1 and D normalizes K, K c ...

1072 i4. L 3 (2) IN THE FSU, AND L 2 (2) WHEN .Cr(G, T) IS EMPTY or Sp 4 (4), or else Lu/0 2 (Lu) ~ SL 3 (4). The first case is ...

14.7. FINISHING L 3 (2) WITH (vG1) ABELIAN 1073 Now F1 is invariant under the symmetry interchanging 'Yl and a, so by this sym- ...

1074 14. L 3 (2) IN THE FSU, AND L 2 (2) WHEN .Cf(G, T) IS EMPTY Suppose first that m(U 0 ) = 2. Again (2) holds. Also T acts on ...

14.7. FINISHING Ls(2) WITH (vGi) ABELIAN 1075 Therefore 11 appears in one of the cases (2)-(13) of F.6.18. Further the subcase o ...

1076 14. Ls(2) IN THE FSU, AND L 2 (2) WHEN Cf(G, T) IS EMPTY 14.7.70.3, VU 0 =Un Ua = [U, Ua]V. Thus Ua centralizes [U, Q], so ...

i4.7. FINISHING Ls(2) WITH (VG1) ABELIAN 1077 8yb(K). That is, (1) holds. Further as 8 :::; 02 (Li) and [f' : 02 (Li)[ = 2, (2) ...

1078 14. L 3 (2) IN THE FSU, AND L 2 (2) WHEN L:r(G, T) IS EMPTY Theorem 14.7.63, U is the £ 2 (4)-module for H*. By 14.7.74.2, ...

14.8. THE QTKE-GROUPS WITH .Cr(G, T) #- 0 1079 Suppose that L / 02 (L) ~ A5. Then the choice of L above was forced, so that K / ...

...

Part 6 The case £1(G, T) empty ...

...

CHAPTER 15 The case £r(G, T) = 0 In this chapter, we complete the treatment of the case £1(G, T) empty. Since the previous chapt ...

io84 i5. THE CASE £,f(G, T) =^0 PROOF. As Mis maximal in M(T) under :Sand V = V(M), we conclude from A.5.7.2 that R = 02(NM(R)), ...

15.1. INITIAL REDUCTIONS WHEN .Cr(G, T) IS EMPTY 1085 that Mc :SM. Then since Mc i. M, A.5.6 gives There is no 1 =f. X::::; V wi ...

1086 15. THE CASE .Cf(G, T) = r/J Thus we may assume that case (i) holds, so MJ = K1 and V =Vo= W1. Thus Mis faithful on W1 and ...