1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

5.2. USING WEAK BN-PAIRS AND THE GREEN BOOK 647 PROOF. Assume that neither (1) nor (2) holds. In particular DL i B as (1) fails. ...

648 5. THE GENERIC CASE: L2(2n) IN .CJ AND n(H) >^1 again K = £ 2. Then B s K = £ 2 , so as S = 02 (BT), a is not the Aut(J2) ...

S.2. USING WEAK BN-PAIRS AND THE GREEN BOOK 649 we obtain p = 3; notice that here L2/02(L 2 ) is not J 1 , since here p = 3 or 5 ...

650 5. THE GENERIC CASE: L2(2n) IN .Ct AND n(H) > 1 PROOF. By construction in Notation 5.1.9, B :::; Na(D). Part (1) holds by ...

5.2. USING WEAK EN-PAIRS AND THE GREEN BOOK 65i (b) a is an extension of the U4(q) amalgam of degree 2 and 02 (KS) is the extens ...

652 5. THE GENERIC CASE: L2(2n) IN .C.1 AND n(H) > i Recall the notion of a completion of an amalgam from Definition F.1.6. L ...

5.2. USING WEAK BN-PAIRS AND THE GREEN BOOK 653 as ILi = IK11, L ~ k, contradicting M = !M(LT). This contradiction completes the ...

654 5. THE GENERIC CASE: L2(2n) IN .Ct AND n(H) >^1 (5) [T: S[ = 2. PROOF. Let M2 := M 2 /U. There is a unique T*-invariant s ...

5.2. USING WEAK BN-PAIRS AND THE GREEN BOOK 655 so CM(X) = Ca(X). Then as INM(X) : CM(X)I = 2 = IAut(X)I, the lemma follows. D R ...

656 5. THE GENERIC CASE: L 2 (2"') IN .Ct AND n(H) >^1 representations of G on r (which is in turn G-isomorphic to the analog ...

5.2. USING WEAK EN-PAIRS AND THE GREEN BOOK 657 as (3(T) = T = ((T), "!(t) E 02 (Ix)t, so"/ E Inn(Ix). Thus adjusting (3 by the ...

658 5. THE GENERIC CASE: L2(2n) IN .Ct AND n(H) > 1 Recall from page 200 of [Asc94] that rr is a base for U if each cycle in ...

5.3. IDENTIFYING RANK 2 LIE-TYPE GROUPS 659 Thus it remains to show Na(S) ::::; Mi. But as SE Syb(Li) and Tis in a unique maxima ...

660 5. THE GENERIC CASE: L2(2n) IN .C.t AND n(H) >^1 Pick z E z# and set Gz := Ca(z). Set Vi := 02(Li) and observe S = V1 V2 ...

5.3. IDENTIFYING RANK 2 LIE-TYPE GROUPS 661 see this leads to a contradiction. Now from the structure of M 1 , E 4 ~ [v, S/Zs] : ...

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CHAPTER 6 Reducing L 2 ( 2n) to n = 2 and V orthogonal In this chapter, we continue our analysis of simple QTKE-groups G for whi ...

664 6. REDUCING L2(2°) TO n = 2 AND V ORTHOGONAL In contrast to the previous chapter, we find now when n(H) = 1 for each H E 7-{ ...

665 M, Hypothesis E.6.1 holds by 6.1.3.l applied to Vi in the role of "V". However as L is transitive on the hyperplanes of Vi, ...

. 666 6. REDUCING L2(2n) TO n = 2 AND V ORTHOGONAL (b) Ki/0 2 (Ki) s=: L~(p), and there is an X-invariant K2 E .C(G, T) n Ki wit ...