1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

667 . Let X := (DL, H). Then XE 'H(T) by 5.1.7.2.iii, as VL is not the S 5 -module. Set U := (zx), Qx := 02(X) and X* := X/Cx(U) ...

668 6. REDUCING L2(2n) TO n = 2 AND V ORTHOGONAL each Li. Choose numbering so that L 0 := Li · · · L; is the product of those fa ...

669 hence Cz(DoT) n [U, L1] is a vector of weight 6, so that Ox* (Cz(D 0 T)) ~Sf,. Now Cz(DoT) = Cz(DL) = Cz(L), so Cx(Cz(DoT)) ...

670 6. REDUCING L2(2n) TO n =^2 AND V ORTHOGONAL LEMMA 6.1.10. (1) r(G, V) ~ n. (2) s(G, V) = m(AutM(V), V) = n. (3) Suppose tha ...

671 {3) Wo(R, V) S CT(V), and if n > 2 then W1(R, V):::; CT(V). (4) Vu:= (VNa(U)/ is elementary abelian, and Vu /U E R 2 (Na( ...

672 6. REDUCING L2(2°) TO n = 2 AND V ORTHOGONAL solvable. This contradiction establishes (5), and so completes the proof of the ...

673 Vis the natural module for L, [S, VJ= Zs; therefore Vis central in SE Syl 2 (Hs), and hence UH~ (VH) E R2(Hs) by B.2.14. Thi ...

674 6. REDUCING L 2 (2n) TO n =^2 AND V ORTHOGONAL. Then applying the subsequent arguments with Fin the role of "B", [F, Vo] =Zs ...

675 Xt 9! 83, 85 or L3(2). In particular now 02,z(P) = 02(P) as the multiplier of these groups is a 2-group. Thus P* = P/02(P). ...

676 6. REDUCING L2(2n) TO n = 2 AND V ORTHOGONAL (3) <P(E) = 1, E/V ~ Z(LifV), and E = keru(Mv) '.'::! Mv. (4) <P(R) ~ E, ...

677 By 6.1.24.2, R =f. 1 ass;::: 1. By 6.1.24.4, R is faithful on F(J). Thus by 6.1.24.3, R* is faithful on E(I*), so there is K ...

678 6. REDUCING L2(2°) TO n =^2 AND V ORTHOGONAL be in the list of F.l.12. As n = 3 and k = 1, the only possibility is the^3 D4( ...

6.2. IDENTIFYING M22 VIA L2(4) ON THE NATURAL MODULE 679 in [Asc86a]. Therefore to prove (2), it remains to show Fu := Cp(u) = 1 ...

680 6. REDUCING L2(2n) TO n = 2 AND V ORTHOGONAL the role of V, Z :=:; VL. Further replacing V by VL if necessary, we may assume ...

6.2. IDENTIFYING M22 VIA L2(4) ON THE NATURAL MODULE 681 field automorphism on Lg /0 2 (Lg) and Vg; and of type 3 if F is of nei ...

682 6. REDUCING L2(2n) TO n = 2 AND V ORTHOGONAL U 0 is also of field type. Also (V; VB) S Ca(Uo), and by 6.2.4.4, VS 02(Ca(Uo)) ...

6.2. IDENTIFYING M22 VIA L2(4) ON THE NATURAL MODULE 683 such that K = [A, K], A S K, K* induces GL(U 1 ) on U 1 := (BK) of rank ...

684 6. REDUCING L2(2n) TO n =^2 AND V ORTHOGONAL Set LK := 02 (NK(RK)), so that LK/02(LK) ~ Z3. First suppose LK:::; M. As K :S ...

6.2. IDENTIFYING M22 VIA L2(4) ON THE NATURAL MODULE 685 As Va i Nc(V(3), Va i G~~), so d(a) = b with a,"(,··· , 'Yl a geodesic, ...

686 6. REDUCING L2(2n) TO n = 2 AND V ORTHOGONAL K* is the direct product of copies of GLm(2) for some m 2: 3. Next as V <I T ...