1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

15.3. THE ELIMINATION OF Mf/CMr(V(Mr)) =Sa wr Z 2 1127 as YL :S M2, 02 (M2) = YL is S-invariant. Hence S also acts on L and Ye, ...

1128 15. THE CASE .Cr(G, T) = f/J PROOF. Observe that 15.3.19, 15.3.20, and 15.3.21 have eliminated all other possibilities for ...

15.3. THE ELIMINATION OF Mr/CMf(V(Mf)) = Ss wr Z2 1129 Set Q := [02(L), L] as in C.1.34, and observe that [Z, L] :::; UL :::; Q. ...

1130 15. THE CASE .Cr(G, T) = 0 holds, since in the other cases in C.1.34 there are at most two such factors. Case ( 4) is elimi ...

lS.3. THE ELIMINATION OF Mr/CMr (V(Mr)) = 83 wr Z 2 1131 By 15.3.26.2, L = [L, Y+J. Comparing the list of Theorem B.5.6 with tha ...

1132 15. THE CASE .Cr(G, T) =^0 CAut(L)(Y+) = 1. Then by 15.3.29.2, V1 S Cs(Y+) S Cs(L), so LS Cc(Vi) SM by 15.3.29.2, contrary ...

i5.3. THE. ELIMINATION OF Mr/CMr(V(Mr)) = Ss wr Z 2 1133 of order 2 for i = 1, 2. Further Zi V2 = Zl/2 = (zY+)::::; (ZL) = U 1 , ...

1134 15. THE CASE .Cf(G, T) =^0 (I)-(IV), and also one of the analogous conclusions on Ul. Then inspecting (I)- (IV), we conclud ...

15.3. THE ELIMINATION OF Mr/CMr(V(Mr)) = Ss wr Z2 1135 Again we will divide the proof into two cases: m 3 (Y+) = 2 and m 3 (Y+) ...

1136 i5. THE CASE .Cf(G, T) = 0 L 3 (2n) with n even. As SE Syb(I) acts on Y+, Y+S acts on the Borel subgroup B of L over Sn L = ...

15.3. THE ELIMINATION OF Mf/CMr(V(Mr)) =Sa wr Z 2 1137 Nc(Vi) = I, so as M1 is transitive on V 1 #, V 1 is a TI-subgroup of G by ...

1138 15. THE CASE .Cr(G, T) = r/J L. Now Y 1 centralizes Vz and hence also VL, whereas [VL, Y1] = [VL, X] # 1, a contradiction. ...

15.3. THE ELIMINATION OF Mf/CMf (V(Mf)) = Ss wr Z 2 1139 By 15.3.42, L+ is of Lie type and characteristic 2, and hence is descri ...

1140 15. THE CASE Cf(G, T) = 0 of 15.3.41, using the same proof, but replacing the appeal to 15.3.36.1 by an appeal to (a): (e) ...

15.3. THE ELIMINATION OF Mf/CMr(V(Mr)) = Ss wr Z2 1141 fused in G. Recall weak closure parameters r(G, V) and w(G, V) from Defin ...

1142 15. THE CASE .Cr(G, T) = 0 (4) 02(H*) = 1. PROOF. By 15.3.46.5, Na(Zs) ::::; M = Na(V), so that part (c) of Hypothesis F.9. ...

15.3. THE ELIMINATION OF Mr/CMf(V(Mr)) = 83 wr Z 2 1143 are equivalent. Further Na(Zs) = NM(Zs) by 15.3.46.5, so as N!VJ(Zs) = f ...

1144 15. THE CASE .Cr(G, T) =^0 Y 0 • Thus L :::; Nc(Yo) = M, contrary to (q) since we just saw 03(L) =J Y 0 . This contradictio ...

15.3. THE ELIMINATION OF Mr/CMf(V(Mf)) = 83 wr Z 2 1145 Observe that L* is A 6 or Ln(2), 3::::; n::::; 5, since in the remaining ...

1146 15. THE CASE .Cf(G, T) = 0 center Zs. Then V_H = (VH) is generated by transvections, tJ = (Zff), and V :SJ T, so by G.6.4.4 ...