1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

Part 2 The treatment of the Generic Case ...

Part 1 has set the stage for the proof of the Main Theorem by supplying infor- mation about the structure of 2-locals, establish ...

CHAPTER 5 The Generic Case: L 2 (2n) in £1 and n(H) > 1 In this chapter we assume the following hypothesis: HYPOTHESIS 5.0.1. ...

630 5. THE GENERIC CASE: L2(2n) IN C.t AND n(H) > 1 an obstacle to applying this recognition theorem: the case where K ~ £* ( ...

5.1. PRELIMINARY ANALYSIS OF THE L 2 (2n) CASE 631 the results of section D.3 rather than those of section 3.2 based on the FSU. ...

632 5. THE GENERIC CASE: £2(2") IN .CJ AND n(H) >^1 (2) n = 2 or 4, [V, L] is the natural module for L, and [Z, H] = 1. (3) n ...

5.1. PRELIMINARY ANALYSIS OF THE L 2 (2n) CASE 633 Also ifs is the involution in (a'), then GA(U) = S(s)D_ and U = (zD+), so U = ...

634 5. THE GENERIC CASE: L2(2n) IN .Ct AND n(H) > 1 permutes {DiZ(8), D 2 Z(8)}. Then 02 (Na(8)) acts on DiZ(8), and indeed c ...

5.1. PRELIMINARY ANALYSIS OF THE L 2 (2n) CASE (1) Sn KE Syb(K). (2) Sn LE Syb(L). (3) If K* is of Lie rank 2, then either (i) S ...

636 5. THE GENERIC CASE: L2(2n) IN .Cf AND n(H) > i PROOF. We may assume D does not act on K, so in particular, D =f. 1. As k ...

S.l. PRELIMINARY ANALYSIS OF THE L 2 (2n) CASE 637 embeddings of As in A.3.14, we conclude K* ~ Ji. As D f:_ Nc(K) is cyclic, we ...

638 5. THE GENERIC CASE: L2(2n) IN .Ct AND n(H) > 1 Suppose first that K is a block. Then so is k., and of the four subgroups ...

5.1. PRELIMINARY ANALYSIS OF THE L 2 (2n) CASE 639 Now as K ~ .Cj(G, T), K centralizes R 2 (KT) by 1.2.10, so that H = KT centra ...

640 5. THE GENERIC CASE: L2(2n) IN .Cf AND n(H) > i composition factor. By F.1.12.II, T:::; K+. But then T:::; K+ S CK, contr ...

5.1. PRELIMINARY ANALYSIS OF THE L 2 (2n) CASE 641 where Wi = V, W2 =Un U^1 , for some l EL - Gi, and W/W 2 is the sum of r natu ...

642 5. THE GENERIC CASE: L2(2n) IN Ct AND n(H) > i This contradiction shows that D'Y < U'Y' Therefore there is /3 E r('Y) ...

5.1. PRELIMINARY ANALYSIS OF THE L 2 (2n) CASE 643 Recall Sn LE Syl2(L), so SDs ::::1 TDs, and hence SoD 5 is a subgroup of G fo ...

644 5. THE GENERIC CASE: L2(2n) IN .CJ AND n(H) >^1 of Theorem 5.1.14 holds. Hence by our remark after 5.1.16, Theorem 5.1.14 ...

5.1. PRELIMINARY ANALYSIS OF THE L 2 (2n) CASE 645 L1(G, T) using 1.2.10, contrary to 5.1.14.1. Thus 5.1.21 says [Vi, QK] = Z 1 ...

646 5. THE GENERIC CASE: L2(2") IN .Ct AND n(H) > i Observe that Hypothesis D.3.1 is satisfied, with YT, Y*, Ur, W in the rol ...