so by (2), V2 ::::; Ca(B) ::::; No(V9). Hence D := Cvg (V2) is of corank at most n + 1
in V9, so from the action of Mfr on V9 either:
(i) Dis of corank exactly n in V9 and V2 induces transvections with axis Don
V9, or
(ii) n = 2, 1 = SL 3 (4), and V2 induces a group of field automorphisms on
AurL9(V^9 ).
By (1), [V 2 , A]::::; VnVg::::; V{' nVj_^9 w for some y EL and w E £9, so A::::; 'hand (i)
holds; then since [V2, A] -f 1, [V2, A] =Vi-As V2 induces transvections with axis D
on V9, Vi = [A, V2] E V{L9
, so we may take g E Gi. But then as [V, V^9 ] -f 1, we
have a contradiction to our assumption that (3) fails. This shows V2 ::::; Ci(T, V),
and completes the proof. 0LEMMA 11.5.4. Ca(Vi) i M, so we may choose H ::::; Gi with 02 (H) ::::;
Ca(Vi).
PROOF. If Li < K, then K ::::; Ca(Vi) but K i M. On the other hand, if
Li = K, then by 11.5.3.3 (V^01 ) is nonabelian, so Gi i M since V::::; Z(02(M)).
Hence Ca(Vi) i M by 11.2.3.2, so using 11.4.3.2 and the argument we made just
before 11.5.2, we can choose H ::::; Gi with 02 (H) ::::; Ca(Vi), while aintaining the
condition 02 (H)::::; Kor [K, 02 (H)]::::; 02 (K). 0Because of 11.5.4, the set Hi := H(LiT, M)nGi is nonempty. In the remainder
of this section we choose Hi E Hi and set UH := (V 3 H^1 ).
LEMMA 11.5.5. (1) UH::::; 02(Hi).
(2) UH::::; Z(02(Hi)), and iP(UH)::::; Vi.
(3) If K =Li, then UH is a direct sum of natural modules for K/02(K).
PROOF. Observe that Hypothesis G.2.1 is satisfied with 113, Hi in the roles of
"V, H"; hence (1) and (2) hold by G.2.2. Further V3 is the natural module for
Li/0 2 (Li); so if Li ~ K, then as K ::::l Gi, (3) holds. 0LEMMA 11.5.6. Let Y := Ca(Vi) n Co(K/02(K)). Then
(1) CIY/, q - 1) = i.
(2) mg(Y) ::::; 1, and if n is even, then Y is a 3'-group.
{3} If IE C(Y) then [I,X]::::; 02(Y).
(4) If P = [P, X] :::; T and il?(P) ::::; 02(Gi), then [P, Y] ::::; 02(Y).(5) [02(LT),X]::::; 02(KT).
(6) If 02 (Hi) ::::; K, then m(A/A n 02 (Hi)) ::::; 1 for each elementary subgroup
A of Ri.
PROOF. For pa prime divisor of q - 1, mp(X) = 2 and Cx(Vi) = X n Li =
X n K, so X n Y = 1. Next Op(X) normalizes Y and hence a Sylow p-group Yp
of Y-so as Gi is an SQTK-group, Yp = 1, proving (1). Similarly as Y centralizes
Li/02(Li) of order divisible by 3, m 3 (Y) ::::; 1, the first requirement of (2).
Assume n is even. Then Y is a 3'-group by (1), completing the proof of (2). Fur-
ther if IE C(Y), then I/0 2 (I) ~ Sz(2k). If (3) fails then by (1), X/Cx(I/0 2 (I))
is a nontrivial group of field automorphisms on I/0 2 (I). Let B be an XT-invariant
Borel subgroup of Io := (IT). Then using 1.2.1.3 as usual, either B = NB(T), or
I< Io and NB(T)T/T ~ Z 2 k_i. In either case, X acts nontrivially on NB(T)T/T.