528 2. CLASSIFYING THE GROUPS WITH IM(T)I = 1
2.4.2, S1 E Syl2(H1) and S1 EU. Then as IH1l2 ::'.": IHl2 = ISi by hypothesis, we
conclude from maximality of 1$1 over U that IS1I = ISi. Then by maximality of
IHl 2 over members of r containing a pair with a member of U of maximal order, we
conclude H1 E I'*, so that H1 E I' 0 in this case as well, completing the proof. D
Since HE I'(), 2.3.8.4 says H = (H n M)L 1 ···Ls, where Li is an L2(2n)-block
with n > 1, an As-block, or an A 5 -block; further Li f:. M, ands :::; 2. Since S, H
play the roles of "U, Hu" in the previous section, in the remainder of this section U
will instead denote the module U(L 1 ) = [0 2 (L 1 ), Ll] in the notation of Definition
C.1.7. Furthermore we set:
L := Ll, Lo := (Ls), and U 0 := (Us).
Then Lo :::l H by 1.2.1.3, so Lo E 1-le by 1.1.3.1, and hence LoS E 1-le. Further
L 0 S f:. M, so (S, L 0 S) E U(L 0 S) and hence L 0 S E re. Then LoS E I' 0 by 2.4.3.2,
so replacing H by L 0 S, we may assume H = L 0 S. Then from section B.6:
H = LoS is a minimal parabolic and L is a Xo-block.
LEMMA 2.4.4. If 1 #So :::; S with So :::l H, then Nr(So) = S.
PROOF. By 2.3.7.2, Na(So) E I'o and S E Syb(Na(So)). In particular S =
Nr(So). D
LEMMA 2.4.5. (1) Hypotheses C.5.1 and C.5.2 are satisfied with S in the roles
of both 'TH, R" for any subgroup Mo ofT with S a proper normal subgroup of Mo.
(2) Assume S:::; Mo:::; T with IMo: SI= 2 and set D := Cs(Lo). Then(a) Q = UoD E A(S).
{b) For each x E Mo - S, 1 = D n nx and U 0 f:. Q.
(3) Assume either that L is an As-block, or that L =Lo is an L 2 (2n)-block oran A5-block. Then the hypotheses of Theorem C.6.1 are satisfied with T, S in the
roles of "A, TH".
(4) Uo = 02(Lo).PROOF. We saw that H = L 0 S is a minimal parabolic, and the rest of Hy-
pothesis C.5.1 is straightforward. As S is proper in Mo, Hypothesis C.5.2 follows
from 2.4.4. Thus (1) holds.
Choose Mo as in (2) and set Do := CBaum(s)(Lo). This is the additional
hypothesis for C.5.6.7; and that result implies (4); and also says that Q = U 0 D 0 ,
Q E A(S), and D 0 nD 0 = 1 for each x E M 0 -S. As Q E A(S), Do= Cs(Lo) = D.
By C.5.5, there exists y E Mo with ui 1:. Q. Then as Uo :::l S and IMo : SI = 2,
Mo - S = { xo E Mo : U 0 ° 1:. Q}, so the proof of (2) is complete.
Finally assume the hypotheses of (3). The first three conditions in the hy-
pothesis of Theorem C.6.1 are immediate, while condition (iv) follows from 2.4.4,
establishing (3). D
LEMMA 2.4.6. Lo ::] GQ.PROOF. By 2.4.3.1, SE Syl2(GQ) with GQ E I' 0. Hence we may apply 2.3.8.4
to GQ, to conclude that GQ is the product of NM(Q) with a product of xo-blocks.
But using 1.2.4 and A.3.12, no larger xo-block contains an S-invariant product Lo
of xo-blocks, so we conclude Lo ::] GQ. D