860 i2. LARGER GROUPS OVER F2 IN .Cj(G, T)
Then as Lo is irreducible on D 0 , and 1-=f. b ~ Loi', we conclude b =Do~ E4.
Next by 12.8.10.6, Cv(U) = (D n Zu)Vf, so using symmetry between U and
U9,
2 = m(D) = m(D/Cv(U)) = m(D/(D n Zu)Vf) = m(Zu/(D n Zu)Vi). (*)
Recall g E h :::; Na(L 0 ). Further b centralizes (D n Zu )Vi; but iJ -=I- 1 does
not centralize Zu since Ki = [Ki,DJ and Ki is nontrivial on Zu. Therefore as
L 0 is irreducible on iJ, and hence on its g-i-conjugate Zu/(D n Zu)Vi, while Lo
normalizes Czu(D), we conclude from(*) that Czu(D) = (DnZu)Vi is of index
4 in Zu. Recall Ki E Cj(G, T), and Vv E R2(KiT). By Theorem 12.6.34, each
Iv E Irr+(Ki, Vv) is a natural 4-dimensional module for KJ.. As Lo is irreducible
on Zu/Czu(D), m(Iv/CrD(D)) = 2, so Zu = Czu(D)Iv and Crn(D) =[Iv, DJ=:
Dr is of order 4. As Zu :::; Ca(Vf) :::; Na(D), Dr :::; D. Further as Ki = [Ki, DJ, Ki
centralizes Zu/Iv, so Iv= [Zu,KiJ. Therefore [Vi,0^2 (Po)J:::; [ViZu,0^2 (Po)] =
Viiv, so Po acts on Cv 2 rn (Lo) = ViCrD (Lo). Therefore if Orn (Lo) = 1, then Po
acts on Vi, contrary to 12.8.13.8.
Thus CrD (Lo) -=f. 1, so since AutLa (Iv) is a minimal parabolic of GL(Iv) and Lo
normalizes b, Orn (Lo)= Dr, and so Po acts on ViDr and on Dr. Finally I2 acts on
V 2 and centralizes Dr by 12.8.10.2, as Dr :::; Zu n Zfj, so Y := (Po, I2) acts on ViDr
and Dr. Then Autr 2 r(V 2 Dr) and Autp 0 (V2Dr) are the two minimal parabolics
of GL(ViDr) ~ L4(2) stabilizing the 2-subspace Dr; in particular, I2Cy(V2Dr) is
normal in Y. But now ash centralizes Dr, Po normalizes [ViDr, hCy(V2Dr)J =Vi,
a case we eliminated in the previous paragraph. This contradiction completes the
proof of 12.9.4. D
By 12.9.4, (VG^1 ) is abelian, and hence (cf. 12.8.6) so is UH = (VH) for each
HE Hz.
LEMMA 12.9.5. (1) Ca 1 (Ki/02(Ki)):::; M, so KiT E Hz.
{2) Ki/02(Ki) ~ A1, L4(2), or Ls(2).
{3) If n = 4 and Ki/02{Ki) ~ L5(2), then Li02(Ki)/02(Ki) is the centralizer
of a transvection in Ki/02(Ki).
PROOF. Observe first that Out(Ki/0 2 (Ki)) is a 2-group for each possibility
in 12.9.3, including Ki =Li, so that Gi = KiTCa 1 (Ki/0 2 (Ki)).
We will combine the proofs of the three parts of the lemma, but in proving (2)
we will assume that (1) has already been proved. Thus when proving (2), Li <Ki
since G1 i M, so that Ki is described in 12.9.3. We consider three cases:
Case I. If (1) fails, pick Hi E H*(T, M) with 02 (H1) :::; Ca 1 (Ki/0 2 (Ki)), and
let H := H1L1.
Case IL If (2) fails, then Li <Ki but Ki/0 2 (K1) is not A1, L 4 (2), or L 5 (2),
and we let H := KiT.
Case III. If (3) fails, then Li02(K1)/0 2 (Ki) is a parabolic determined by an
end node and the adjacent node in the Dynkin diagram for L 5 (2), and we pick
H1 E H*(T, M) to be the minimal parabolic of K 1 determined by the remaining
end node, and let H := HiLi.
In each case H E Hz. As 12.9.4 provides condition (2) of 12.8.6, the latter
result says that H satisfies Hypotheses F.8.1 and F.9.8 with Vin the role of "V+".
Thus· we may apply the results in sections F.8 and F.9. In particular we adopt the
notation of sections F.7 and F.8 (or F.9) for the amalgam generated by Hand LT.