1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

5.3. IDENTIFYING RANK 2 LIE-TYPE GROUPS 661

see this leads to a contradiction. Now from the structure of M 1 , E 4 ~ [v, S/Zs] ::;

(Vz n S)/Zs, so m(Vz) ~ 4. Therefore as E 16 ~ Q = J(T), we must have Vz = Q.

Next as CT(Q)::; Q, Gz/Q::; Out(Q) ~ ot(2), so IGz: Tl= 3 or 9. As

IGz : Tl ~ IZ~z I ~ m(Vz) = 4,

IGz : Tl = 9. As m2(Q) = 3 and m(V'i) = 4, Vi i:. Q; indeed [Vi, v]Zs ::; Q and

[Vi, Q] ::; 1/i, so that ViQ/Q has order 2 and induces an involution of type a 2 on Q, so

it centralizes a nontrivial element in 02 (Gz/Q) ~ E 9. Therefore 02 (NcJViQ)) i=

1. However by 5.3.6.1, Vi is weakly closed in ViQ; so 02 (Nc,(ViQ)) ::; 02 (Gz n

Mi) = 1, contradicting the previous remark. This contradiction completes the proof

of 5.3.7. D

Observe that 5.3. 7 contradicts our assumption that 5.3.4.2 holds. So the proof

of Theorem 5.2.3 is complete.