1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

4.1. SOME GENERAL MACHINERY FOR PUSHING UP 607 Set Mi := NM_ (R), and pick R+ E Syl2(GM(M+/02(M+)) so that Ri := NR+(R) E Syl2(C ...

608 4. PUSHING UP IN QTKE-GROUPS PROOF. Notice that the pair I,R satisfies the hypotheses of 4.1.4 for any HE H(I, M). Since IEμ ...

4.2. PUSHING UP IN THE FUNDAMENTAL SETUP 609 (1) holds, and NH(V) s MH and V = [V, Mo] by 4.1.5. As V s Z(R+) and 02(MoR) s R+, ...

610 4. PUSHING UP IN QTKE-GROUPS By A.4.4.3, a 2 ,F* (M) 'f:_ H, so in view of 4.2.9, there is K E C(M) with K/a 2 (K) quasisimp ...

4.2. PUSHING UP IN THE FUNDAMENTAL SETUP 611 We now appeal to Theorem C.4.8. By Theorem C.4.8, LH ::::l MH, so L = Lo ::::l M si ...

612 4. PUSHING UP IN QTKE-GROUPS of groups of type Ru in chapter J of Volume I, Ki := Ki/Q1 ~ 85, and from J.2.3, CQ 1 (X 1 ) ~ ...

613 4.3. Pushing up L 2 (2n) In the first exceptional case of Theorem 4.2.13 where L :SI Mand L/0 2 (L) ~ L2(2n) for n > 1, i ...

614 4. PUSHING UP IN QTKE-GROUPS PROOF. As I= LR::; LS, maximality of S implies (a). Then as S < Nr(S), (a) and 4.3.1 imply N ...

615 particular Kz/02,F(Kz) is not SL2(P) for any odd prime p. This rules out cases (c) and (d) in 1.2.1.4, so that Kz/02(Kz) is ...

616 4. PUSHING UP IN QTKE-GROUPS to Ki. M. Hence K = [K, Ut]. Recall also that V = [fh(Z(R+)), L] is T-invariant, so V =Vt. As R ...

6i7 that A(S) = {U,A} is of order 2 with V =Un A of rank 4n. We now obtain a contradiction similar to that in the L 3 (2n)-case ...

618 4. PUSHING UP IN QTKE-GROUPS Na(R2) = NH(R 2 ) acts on the parabolic K 2 of K, since we saw after 4.3.6 that K :::;I H, so ( ...

4.4. CONTROLLING SUITABLE ODD LOCALS 619 K2/02(K2), contrary to CR 2 (X)/CR 2 (K2X) the natural module for K 2 /0 2 (K 2 ). This ...

620 4. PUSHING UP IN QTKE-GROUPS PROOF. Let X := T+B, Q := 02 (X) and X := X/Q. Then F(X) is of odd order, so as B is an abelian ...

4.4. CONTROLLING SUITABLE ODD LOCALS 62i LEMMA 4.4.9. If K is a component of GB, then !KGB I S 2, and in case of equality, K ~ L ...

S22 0 PUSHING UP IN QTKE-GROUPS we chose TB E Syl 2 (MB), TB is transitive on each orbit of MB on parabolics of K containing T ...

4.4. CONTROLLING SUITABLE ODD LOCALS 623 In cases (d)-(f), VB is a natural module for LB/0 2 (LB), so that subcase (i) of case ( ...

624 4. PUSHING UP IN QTKE-GROUPS particular, Mo is 2-closed, and a Hall 2'-subgroup B of Mo is abelian of p-rank at most 2 for e ...

4.4. CONTROLLING SUITABLE ODD LOCALS 625 Assume (1) fails. Then B := CBH(M+/02(M+)) =/:-1. Observe that we have the hypotheses o ...

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