1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

i3.5. THE TREATMENT OF A5 AND A 6 WHEN (v;^1 ) IS NONABELIAN 92i

V x D, we conclude Q E !32(KDD), and in particular Q contains 02 (KDD) by

C.2.1.2. Then as V n 02(KDD) = 1, D = 02(KDD). As GcM(D)(Q) = Q,

KTo = GM(D). Hence K+Tt ~ KTif Dis the 2-local NK+(Q+) D = NK+(V+) D in

the quasisimple group Kfi. But inspecting the groups in Theorem C (A.2.3), we

find no such 2-local. This contradiction establishes the claim that Gr(K) = 1.

As Gr(K) = 1, V = Q = 02(LT), so 02(M) = V and M =LT by 3.2.11. Thus

(7) holds, and hence also (8) and (9) by an earlier observation. Thus it remains to

establish (6).

By A.1.6, Qi := 02(Gi) S QH. Also Qi S 02(LiT), and by (8) and (9),

either 02(LiT) = 02(Li)V = UHV, or L/V ~ 85 and 02(LiT) = UHVGr(Li),

with Gr(Li) of order 4. Next UHV n QH =UH and as HS Gi, UH S Ua 1 S Qi.

We conclude that Qi = UH or UHGr(Li). In either case, Qi = UHZi where

Zi := Z(Qi) is of order at most 4, and <I>(Qi) =Vi. Thus Gi preserves the usual

symplectic form on Qi := Qi/Zi. Now m(Qi) = 4 as UH ~ Q~, So Gi/Qi S

8p(Qi) ~ 85. Then as T S H and H/Qi ~ 83 x 83 or 83 x Z3, it follows that

Gi = H. Thus Li ::9 Gi. Finally for any Hi E 1iz, as Li S Hi S Gi and Li ::9 Gi,

Li :::;I Hi; so by symmetry between Hand Hi, Hi = G 1. This completes the proof

of (6), and hence of the lemma. D

We can "now proceed to the identification of the groups in Theorem 13.5.12,

under the assumption that Li is normal in H.

PROPOSITION 13.5.16. If L/02(L) ~ A5 and Li :::;I H, then G ~ U4(3).

PROOF. By 13.5.15.6, H := Gi is the unique member of1iz. Let U :=UH and

y E L 2 - T, so that U ~ Q~ by 13.5.15.5. We consider the two cases of 13.5.15.9.

Suppose first that M = L. Then 02(Gi) = U by 13.5.15.9, and Un UY= \12.

Hence G is of type U 4 (3) in the sense of section 45 (page 244) of [Asc94], so by

45.11 in [Asc94], G ~ U4(3).

Thus we may assume that M/V ~ 86 ; in this case we will obtain a contradiction

using transfer, eliminating shadows of extensions of U4(3). By 13.5.15.9, Z(QH) =

Gr(Li) is of order 4.

Let TL := T n L E 8yl2(L), and define I as in the proof of 13.5.15. From

the proof of 13.5.15, U = 02(Li) = 02(I), VUE 8yh(I), and [Li, I] S U. Then

VU= V0 2 (Li) S TL E 8yl2(L), so TL E 8yb(TLILi), Now Lis transitive on V#,

while ILi is transitive on the involutions in U - Vi, and all involutions in L are

fused into U under L, so we conclude all involutions in TL are in z^0.

Suppose that QH is not weakly closed in H with respect to G. Observe that

Vi = (QH), so Na(QH) = H. Then by A.1.13 there is x E G with QH -=f. Q'H