i4.7. FINISHING L 3 (2) WITH (VG1) ABELIAN 1063over F4, we may apply (1) to Hi to obtain a contradiction. In the second case
Ki = 0
31(Gi) by A.3.18, so Li :S; Ki. Then as K = 02 (NK 1 (K)), Li :::;; K, and
(2b) holds. DLEMMA 14.7.46. Li :S; K.
PROOF. Assume Li i. K; then Hand its action on UH are described in 14.7.42,
and KE C(Gi) by 14.7.45.2. Let B be the Borel subgroup of K containing T n K.
Then BT= CK(V2) from the module structure in 14.7.42, so B normalizes J 2 by14.7.4.2. Further Li '.SI H since case (1) of 14.7.41 holds, so B also centralizes
- -L
V = (V 2 1 ). By 14.7.4.2, I2/02(I2) ~ 83. Set Go:= (hK,T).
· Suppose first that 02(Go) = 1. Then Hypothesis F.1.1 is satisfied with K, h,
T in the roles of "Li, L2, 8", so /3 := (KT, BT, I 2 BT) is a weak EN-pair of rank
2 by F.1.9. Further T '.SI TN1 2 (T n I2), so /3 is described in F.1.12. Then as KT
centralizes Vi with KT/02(KT) ~ 85, and J 2 T/0 2 (hT) ~ 83 , it follows that f3is parabolic isomorphic to the Aut(J2)-amalgam. This is impossible, since in that
amalgam, 02(KT) ~ QsDs while UH:::;; 02(KT) is of 2-rank 9 by 14.7.42.
Thus Go E H(T), so K:::;; Ko E C(Go) by 1.2.4. If K = K 0 , then L 2 = 02 (1 2 )
acts on K by 1.2.1.3, so LT = (LiT, L3) acts on K; then as M = !M(LT),
K:::;; Na(K):::;; M, contrary to 14.7.30. Thus K <Ko, so since Li i. K, Ko i. Giby 14.7.45.2. Then Ko E C1(G, T), so that Ko/02(K 0 ) ~ A 5 or L 3 (2) by 14.3.4.1,
contrary to A.3.14. DWe are now in a position to complete the proof of Theorem 14.7.40.
By 14.7.46, Li :::;; K, so H = KLiT = KT. Further LiT/0 2 (LiT) ~ 83.
Therefore H* ~ 85 by 14.7.45.1.As Li :::;; K, V = [V,Li] :::;; [UH,K], so UH = [UH,K]. Suppose UH E
Irr+(K,UH)· As Vis an LiT-invariant line in UH, UH is not the A5-module.
Then UH/CuH(K) is the L 2 (4)-module, and hence Theorem 14.7.40 holds in this
case.Thus we may assume UH ¢:. Irr+ ( K, UH), and it remains to derive a contra-
diction. By Notation 14.7.1, U~ :::;; Ri with Ri Sylow in K*. Further m(U~) =:
k = 1 or 2, U~ E Q(H*,UH), and k 2 m(UH/DH) by choice of "Yin 14.7.1. As
U~ :::;; Ri :::;; K*, m(W/Cw(U~)) 2 2 for each noncentral chief factor W for Kon
UH, and as UH ¢Irr +(K, UH), there are at least two such chief factors. On theother hand, as U~ E Q(H*,UH), 2k 2 m(UH/CuH(Ua)), so we conclude k = 2,
and there are exactly two noncentral chief factors, both L 2 (4)-modules. Further
2m(U;) = m(U/CoH(Ua)) so by 14.5.18.2, m(UH/DH) = 2, and u; acts as a
group of transvections on DH with center Ai. This is impossible as UH has two
L 2 ( 4 )-chief factors.
Thus Theorem 14. 7.40 is at last established.
LEMMA 14.7.47. Let KE C(H). Then
(1) Li:::;; K, and
(2) K/02(K) ~ Ln(2) or An for suitable n, or G3(2)'.
PROOF. As KTLi E Hz by 14.7.32, we may take H = KTLi. By 14.7.36,
K/0 2 ,z(K) is not sporadic, so by 14.7.30 we may apply F.9.18.4 to conclude that
either K/0 2 ,z(K) is of Lie type in characteristic 2, or K/02(K) ~ A1. Assume the
first case holds. If K/0 2 ,z(K) is not defined over F3, then H* ~ 85 by Theorem