1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

10.2. WEAK CLOSURE PARAMETERS AND CONTROL OF CENTRALIZERS 747 CE(BK) = CE(Di) = E2 are of rank n. Thus m = n by (*), and D 1 = B ...

748 10. THE CASE LE .Cj(G, T) NOT NORMAL IN M. It will suffice to show Na(T 1 ) acts on Xi for i = 1and2, since then Na(T1) acts ...

10.2. WEAK CLOSURE PARAMETERS AND CONTROL OF CENTRALIZERS 749 that mp(Ki) S 1 for each p E 7r(ILI) to the list of embeddings of ...

750 10. THE CASE LE .Cj(G,T) NOT NORMAL IN M. By B.4.8.2, A= R 1 and rz Q, A= 1, so by B.4.8.4, Zq = V1CzQ(L). This shows ZQ = V ...

10.2. WEAK CLOSURE PARAMETERS AND CONTROL OF CENTRALIZERS 7Sl the other hand if I~ Sp4(2n), then Lis indecomposable on 02 (L), s ...

752 10. THE CASE LE Cj(G, T) NOT NORMAL IN M. A 7 -block, or there is a nontrivial characteristic subgroup C of 8 normal in ITv. ...

10.2. WEAK CLOSURE PARAMETERS AND CONTROL OF CENTRALIZERS 753 as It= 0 31 (Ca(u)) since I= 0 31 (H). Then CrTv(E) = C1•TJE), so ...

754 10. THE CASE LE .Cj(G, T) NOT NORMAL IN M. Assume (3) fails. If V centralizes Vz, then as Caz (Vz) :::; M by (1), V :::; 02 ...

10.3. THE FINAL CONTRADICTION 755 Thus k ~ m-a, so by paragraph two, case (3) of 10.l.l holds with w( G, V) = k = l. Hence V f:. ...

756 10. THE CASE LE .Cj(G, T) NOT NORMAL IN M .. W 0 ::::; 02 (H) by B.6.8.3d, so Wo = Wo(02(H), V) and then HS Nc(Wo) SM, contr ...

10.3. THE FINAL CONTRADICTION 757 L2(2m) with m 2 2. However the embedding K < Iz does not occur in the list of A.3.14, so we ...

...

CHAPTER 11 In this chapter, we complete the elimination of the groups possessing a pair L, V arising in the Fundamental Setup (3 ...

PROOF. By Theorem 7.0.1, Vis an FF-module for AutaL(V)(L). By construc- tion in the FSU, V = (V!) for some Vo E Irr +(L, R 2 (LT ...

11.1. THE SUBGROUPS Na(V;) FOR T-INVARIANT SUBSPACES V; OF V 76i that Na(Vi) E 7-le by 1.1.4.6. Notice when L ~ SL 3 (q) that V3 ...

76Z 11. ELIMINATION OF Ls(2°), Sp4(2°), AND G2(2°) FOR n > 1 Next Li = L'f S Ca(Vi) = Gi, so K = [K, Li] S Gi. As Xi is faith ...

11.1. THE SUBGROUPS Na(V;) FORT-INVARIANT SUBSPACES V; OF V 763 so m3(I) = 2. Also .C(G, T) n I contains two members £ 2 , Ko wi ...

natural. Therefore by B.4.14, Wis the adjoint module for K/02(K) and Cw(Xi) is indecomposable of F 4 -dimension 4 for L+/0 2 (L+ ...

ii.i. THE SUBGROUPS Na(V1) FORT-INVARIANT SUBSPACES V 1 OF V 765 (c) q = 4, K/Ooo(K) ~ L2(5), and 02(K) < 000 (K) centralizes ...

m(Q/Q 1 ) ·:::::; m(H^1 (L, V)), so H^1 (L, V) -=/= 0. Therefore by I.1.6, L is an Sp4(4)- block and m(Q/Q 1 ):::::; 2, and by I ...