13.8. FINISHING THE TREATMENT OF A 6 953u;, E Q(H*,UH), so in particular u;, acts quadratically on UH, and hence it fol-
lows from the facts that n = 2, u; is a 4-group, and m(UH/GuH(Ur )) = 4. Now
by F.9.16.2, m(UH/DH) = 2, which is impossible as [DH,u;J = .A. 1 by F.9.13.6,
whereas no 4-group in K 0 induces a group of transvections on a subspace of co9.i-
mension 2 in UH of dimension 8.
It remains to eliminate the case K* ~ Sz(2n). Since B ::=:; M, [B, L 1 ] ::=:;
Lin B ::::; 02(L1), so Li centralizes K 0. However case (a) or (bl) holds, so that
U = U1 EB U2 with Ui the natural module for K*; then EndK· (Ui) = F 2 n with n
odd, and hence [U, Li] = 1. This is a contradiction, since L 1 :::;! Hand V 3 = (V 3 , L 1 ],so UH= [UH,L1].
Essentially the same argument establishes (2): We conclude from parts (4)
and (7) ofF.9.18 that [UH,KJ/G[u~,K](K) is the natural module for K/0 2 (K) ~Sz(2n). Again Li centralizes K* and then also [UH, K], for the same contradiction.
D
By 13.8.16 and F.9.18.4:LEMMA 13.8.17. K* ~ L2(2n), (S)L3(2n)e, Sp4(2n)', G2(2n)', L4(2), L5(2),
A1, A5, M22, or M22-In the remainder of the section, we successively eliminate the cases listed in
13.8.17.
Observe that the second case of 13.8.8 holds, unless K* is one of the groups
A5, A1, or L4(2) allowed by 13.8.5.2 in the first case.
LEMMA 13.8.18. If H = KL 1 T, then
(1) G'Y = (F^0 'Y)G'Ym-i for each F ::=:;UH with Fi. DH.(2) In case (2) of 13.8.8, the hypotheses of 13.8.11 are satisfied.
(3) If case (2) of 13.8.8 holds and UH does not induce a transvection on U'Y,
then m(V .. ;) > 1.
(4) If no member of H* induces a transvection on UH, then m(V,.;) > 1.PROOF. By F.9.13.2 UH ::::; 02(G'Y,'Yb-J, while as H = KL1T, for gb with
bo,'Y1)gb = bb-1,'Y) we have G'Y = K^9 bG'Ym-i· Thus if Fi. DH, then K9b =
[K9b, F], so (1) holds. In case (2) of 13.8.8, DH < UH, so (2) follows by an
application of (1) with UH in the role of "F". Finally 13.8.11.2 and (2) imply (3),
and 13.8.8 and (3) imply (4). DLEMMA 13.8.19. H* is not L3(2).
PROOF. Assume H ~ L 3 (2). Then LJ'T is a maximal parabolic of H*; let
P be the remaining maximal parabolic of H containing T. Since UH = (Vl)
with L 1 T inducing 83 on V3, H.6.5 says UH is one of the following: the natural
module W in which P stabilizes a point, the core U2 of the permutation module
on H/P, the Steinberg module S, WEBS, or U2 EB S. By 13.7.6.1, UH is not
natural. Then since U';,_ E Q(H*, UH) by 13.8.8, it follows using B.5.1 and B.4.5
that UH = U2. By 13.8.8, v; ::::; Ri, so as Ri is not quadratic on UH = U2, it
follows that m(VC:) = 1. This contradicts 13.8.18.4 in view of G.6.4. D
LEMMA 13.8.20. K* is not of Lie type over F2n for any n > 1..