1116 i5. THE CASE .C.r(G, T) =^0Xis described in C.1.34, then Xis an L 3 (2)-block. This completes our preliminary
reductions.
Suppose first that conclusion (i) of 15.2.18.3 holds. Then by 15.2.19.1 and
C.1.28, I= MrLi ···Ls, where Li is ax-block. But then Mr= I by the previous
paragraph, a contradiction.
Therefore conclusion (ii) of 15.2.18.3 holds. Set R := 02(Nr). Then case
(2) of 15.2.19 holds, so Hypothesis C.2.3 is satisfied by I, R, Nr in the roles of
"H, R, MH". If 02,F(J) i Nr, then by C.2.6, there is an A3-block X of I with
X i Nr, contrary to an earlier reduction. Thus 02,F(J) :S Nr. On the otherhand, if 0 2 ,F* (I) :::; Nr, then R = 02 (J) by A.4.4.1, contradicting Nr = Nr ( R) and
Ki Nr. Thus there is XE C(J) with X/0 2 (X) quasisimple and Xi Nr. Further
K = 0^2 (K) normalizes X by 1.2.1.3.
If R does not act on X, then X is a x-block by C.2.4, contrary to an earlier
reduction. Thus R acts on X, so Xis described in C.2.7.3. Let Mx :=Mn X,
and this time set Mo:= Mr n X, so that Mi:= Cx(Zs):::; Mo by 15.2.12.3.
Suppose that case (g) of C.2.7.3 holds. Then X/02(X) ~ SL3(2n), Mx is a
maximal parabolic of X, and (XR,R) is an MS-pair described in C.1.34. Assume
first that n > 1. Then Mi = P^00 , where P is the maximal parabolic of XS
over S other than Mx. As m 3 (KCx(Zs)) = 2, Ko := 02 (K n Mi) =I l; then
as S acts on Ko, n is even. Then 02 ,z(X) > 02 (X), so as m3(K0 2 ,z(X)) = 2,
02 ,z(X) :::; K. This is impossible, as 02 ,z(X) :::; Mx while Kn M = 02(K).
Therefore n = 1, so by an earlier reduction, Xis an L3(2)-block and Mo = Mi
is the maximal parabolic of X over Sn X other than Mx. If X < (Xs) =: X 0 ,
then MxS/02(MxS) ~ ot(2). This is impossible, as MxS ~ S 3 since conclusion
(ii) of 15.2.18.3 holds, while ot(2) has no such quotient group. Thus X ::::! I by
1.2.1.3, so by minimality of I, I= MrX. Then Sis not irreducible on K/0 2 (K)
since m3(KnX) = 1, so Kc:::; Mr by 15.2.21.1. Further KS= HiH 2 by 15.2.21.1,
so
(*) K1 and K2 are the S-invariant subgroups K+ of K with IK+ : 02(K+)I = 3.
As m3(KX) = 2 = m 3 (K), by() we may assume Ki = 0^2 (K n X). Then by
another application of(), [X, K 2 ] :::; 02 (X). Further Ki = 02 (M 0 ) ::::! Mr, so that
K =Kc by 15.2.12.2. Thus CM(V) is a 3'-group by 15.2.21.2, so 0
31
(Nr) =:Yi=
0
31
(Mx ), and Yi is S-invariant with YiS/02(YiS) ~ S3.· As [X, Kz] :S 02(X),
[Yi, K2] :::; K2 n Yi, contrary to 15.2.22.Thus case (g) of C.2. 7.3 is eliminated. An earlier reduction showed that X is
not ax-block; this eliminates case (a) of C.2.7.3, and the subcases of (b) where X
is a x-block. In the remaining cases, m 3 (X) = 2, and then X = B(I) by A.3.18;
so as KS :S Cc(Zs) by 15.2.11.2, KS :::; Mi :S Mo. In particular m3(Mo) = 2,
with KS/02(KS) ~ S3 x 83; so by inspection of the list in C.2.7.3 (recalling that
Out(Sp4(4)) is cyclic; cf. 16.1.4 and its underlying reference), either Xis an A 7 -
block, or X/02(X) ~ L 4 (2) or L 5 (2). The former case was eliminated earlier, so the
latter holds. Now Mi is a proper parabolic of X containing K, so either MiS =KS
is determined by a pair of non-adjacent nodes, or X/0 2 (X) ~ L 5 (2) and MiS is
a maximal parabolic determined by all the nodes except one interior node. Let
U := [(Zff),X]. By B.2.14, (Zff) = UCcz:g)(X), so that Cx(U n Zs)= Cx(Zs).
Now by C.2.7.2, U is an FF-module for (XS)+ := XS/0 2 (XS), and hence is