616 4. PUSHING UP IN QTKE-GROUPSto Ki. M. Hence K = [K, Ut]. Recall also that V = [fh(Z(R+)), L] is T-invariant,
so V =Vt. As R = 02 (LS) by 4.3.4, while SE Syh(H) by 4.3.3.b, 02(KR)::; R.Suppose first that K is an Sp4(4)-block. Then ZK := Cu(K)::; V using I.2.3.3,
and U/V is the natural L 2 (4)-module for L/0 2 (L). So as V =Vt, ut /Vis also the
natural module, with Cu(K)::; V::; UnUt < U, impossible as 02 (L)02(K)/02(K)is a non-split extension of a trivial submodule by a natural module, so that there
is no natural £-submodule.
Suppose next that K is an A 7 -block. Then by C.4.1.1, S induces a transposition
on L/0 2 (L), so that LS/02(LS) ~ S5 = Aut(L/02(L)), and hence T = SR+.
Hence as R::; S < T, R < R+ and so R < NR+(R). We claim that K, R, S,
R+, KS satisfy Hypothesis C.6.2, and the hypotheses of C.6.4, in the roles of "L,R, TH, A, H". Most requirements are either immediate or have been established
earlier-except possibly for C.6.2 and C.6.4.II (recall the latter result uses C.6.3
and in particular verifies its hypotheses), which we now verify: If 1 =f=. Ro ::; R
satisfies Ro :::;I KS, then by 4.3.3.a, Nr(R 0 ) =Sas Ki. M; so by 4.3.4 NR+ (Ro)=R < N R+ ( R), completing the verification of those hypotheses. As T = SR+, we
conclude from C.6.4.10 that e 1 ,2 E Z(T). Then as ei,2 centralizes L, Co(e1,2) ::;
M = !M(LT). Now v := e 3 ,4 is in V, and there is k E K with e~, 2 = v. Then
R+::; Co(V)::; Ca(v)::; Mk, soR+ actsonLk. ButthenR+ actsonK= (L,Lk),
so T =SR+ ::; Na(K) = H, which we saw earlier is not the case.Therefore K is an SL 3 (2n)-block. Thus case (3) of C.4.1 holds, so L is the
stabilizer of the line V of U, so that [U, L] = V. Therefore as t acts on V and
L, also [Ut, L] = V ::; U. This is impossible as we saw K = [K, ut], whereas
K / 02 ( K) admits no involutory automorphism centralizing L0 2 ( K) / 02 ( K). This
contradiction completes the proof of 4.3. 7. DLEMMA 4.3.8. K/0 2 (K) ~ SL 3 (2n), (KR, R) is an MS-pair described in one
of cases (2)-(4) of Theorem C.1.34, and SE Syh(H).PROOF. Recall that K is described in one of cases (1)-(3) of Theorem C.4.1.
As L/02(L) ~ L2(2n) and K is not a block by 4.3.7, conclusion (3) of C.4.1 holds,
so that K/02(K) ~ SL 3 (2n), and one of cases (1)-(4) of C.l.34 holds. Further
4.3.7 rules out case (1). where K is an SL 3 (2n)-block. By 4.3.3, SE Syb(H). DLEMMA 4.3.9. Cs(K) = 1.
PROOF. Let U := fh(Z(02(KS))); as K :::;JH, [U,K]::; 02 (K). By 4.3.8, K is
described in one of cases (2)-(4) of C.1.34, so that [U, K] is the sum of one or two
isomorphic natural modules for K/0 2 (K). So as the natural module has trivial 1-
cohomology by I.1.6 since n > 1, we conclude that U = Cu(K) EB [U, K]. Further L
stabilizes an F2n-line in the natural summands of [U, L] by C.4.1, so C[u,K] (L) = 0.Thus Cu(K) = Cu(L), so Cz(L) = Cz(K), where Z := fh(Z(S)). But Nr(S)
normalizes Cz(L), so if Cz(L) =f=. 1 then No(Cz(L)) ::; M by 4.3.5. Therefore as
Cz(K) = Cz(L) and Ki. M, Cz(K) = 1, establishing the lemma. D
LEMMA 4.3.10. K satisfies conclusion (3) of Theorem C.1.34.
PROOF. By 4.3.8, one of conclusions (2)-(4) of Theorem C.1.34 holds, and asCs(K) = 1by4.3.9, conclusion (4) does not hold. Thus we may assume conclusion
(2) holds, and it remains to derive a contradiction. Then U = 02 (K) is the sum of