1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

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) =Vi, and CA(V^9 ) has rank 4,3, with CA(L) of rank 1,2,

respectively. So in any case m(V9 /CA(L)) = 3 < r, and hence we can continue the

argument in the proof of 9.4.4 to get U 1 Uf :S Af9, and obtain the same contradic-

tion. D

Observe that by 9.4.4, 9.4.5, and E.3.16, Na(Wo) :SM 2: Ca(C1(T, V)). As
m(M, V) 2: 2, s(G, V) 2: 2 by 9.4.3. Then as n(H) = 1, E.3.19 forces H :SM, a

contradiction. This contradiction finally shows that case (1) of 3.2.6 cannot occur,

and hence completes the proof of Theorem 7.0.1 begun in chapter 7.