1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

8.3. ELIMINATING La(2) I 2 ON 9 LEMMA 8.3.13. If V9 n v n z^0 -:/:-0, then V9 ::::; Ca(V). PROOF. This is a consequence of 8.3.1 ...

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CHAPTER 9 Eliminating nt(2n) on its orthogonal module The results in chapters 7 and 8 almost suffice to establish Theorem 7.0.1, ...

730 9. ELIMINATING nt(2n) ON ITS ORTHOGONAL MODULE 9.2. Reducing to n = Our first goal is to show that n :::; 2. We cannot use t ...

9.2. REDUCING TO n = (^2 731) PROOF. Recall that H centralizes Z. By 9.2.4.1, k := n(H) ;::: n - 1, so either k > 1 or n = 2. ...

732 9. ELIMINATING n:tc2n) ON ITS ORTHOGONAL MODULE n = 2, so the lemma holds. Finally if t-=/:-er, then t induces an automorphi ...

9.3. REDUCING TO n(H) = (^1 733) for KX:T*. Comparing the list in A.3.14 with the list of FF-modules in B.5.6, we conclude KX: ~ ...

734 9. ELIMINATING nt(2n) ON ITS ORTHOGONAL MODULE PROOF. By 9.2.7, B = CD(Vi) SK is diagonally embedded in LLt, so D = B(DnL) s ...

9.4. ELIMINATING n(H) = 1 735 9.4. Eliminating n(H) = 1 As we just showed n(H) =f. 2, n(H) = 1 for all H E H*(T, M) by 9.2.5. Th ...

736 9. ELIMINATING nt(zn) ON ITS ORTHOGONAL MODULE XE B(Gv,Tv)· Now Tacts on V1VN and there is g E NL 0 (V1VN) with vg t/:. VN a ...

9.4. ELIMINATING n(H) = 1 737 Thus A :S TL, so A has rank 3 or 4. We now argue as in the proof of 9.4.4: First Cv(A) = Cv(TL) =V ...

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Part 4 Pairs in the FSU over F 2 n for n > 1. ...

In part 4, we prove two theorems about pairs L, V in the Fundamental Setup (3.2.1): In chapter 10, we show that L = L 0. Then in ...

CHAPTER 10 The case L E £j ( G, T) not normal in M. In this chapter we prove: THEOREM 10.0.1. Assume G is a simple QTKE-group, T ...

742 10. THE CASE LE .Cj(G, T) NOT NORMAL IN M. . (2) J(T) ::; NT(L) =Ti. (3) CT(V) = 02 (L 0 T) except' in case (6) of 10.1.1, w ...

i0.2. WEAK CLOSURE PARAMETERS AND CONTROL OF CENTRALIZERS 743 8i,i x 8i,2 with 8i,j 9'! Ds, and T acts transitively as D 8 on th ...

744 10. THE CASE L E .Cj(G, T) NOT NORMAL IN M. LEMMA 10.2.2. Assume HE 1-l*(T, M) with [Z, HJ =f 1, and set W := (ZH). Then (1) ...

10.2. WEAK CLOSURE PARAMETERS AND CONTROL OF CENTRALIZERS 745 (4) Case {1} of 10.1.1 does not hold; that is, n(H) = 1 for each H ...

746 io. THE CASE LE .Cj(G,T) NOT NORMAL IN M. and [Z, K] # 1, also I= [I, J(T)] and 1 # U := [Z, J] E R2(I) using B.2.14. Thus J ...