4.2. PUSHING UP IN THE FUNDAMENTAL SETUP 611We now appeal to Theorem C.4.8. By Theorem C.4.8, LH ::::l MH, so L =
Lo ::::l M since LT = LI. As 02,F• (H) i M by 4.2.5, one of cases (1)-(9)
of Theorem C.4.8 holds. By Theorem C.4.8, LH :::; K E C(H) with K i M
and K/02(K) quasisimple. As L/0 2 (L) is quasisimple, fh(Z(R+)) = R 2 (LT), so
V+ = [R 2 (LT), L]. This completes the proof of (2). []Now we come to a fundamental result, showing that many subgroups of LT
covering L/02(L) are uniqueness subgroups, whenever Vis not on a short list of
FF-modules.THEOREM 4.2.13. Assume Hypothesis 4.2.1 and let I E I. Then either M =
!M(I); or L ::::l M, V := [R2(LT),L] is an FF-module for LT/0 2 (LT), and oneof the following holds:
(1) L/02(L) ~ L2(2n).
(2) L/02(L) ~ L3(2) or L4(2), and V/Cv(L) is either the sum of isomorphic
natural modules, or the 6-dimensional orthogonal module for L4(2). ·
(3) 02 (I n L) is an A5-block or an exceptional A1-block.
(4) 02 (InL) is a block of type A5, and for each z E Cv(T)#, Vi 02(Ca(z)).
(5) 02 (I n L) is a block of type G 2 (2), and if m(V) = 6 and Vs is the (T n !)-invariant subspace of V of rank 3, then Ca(Vi) i M.
PROOF. Assume I EI, H E M(I) - {M}, and set R := 02 (J). Since T E
Syb(G), we may assume that R:::; T n IE Syb(I). Define M_ as in 4.2.12; by
4.2.12.1, IE μ.
Let I :S Ii E μ. Then I1 EI, and if Ii satisfies one of the conclusions (1)-(5)
of the Theorem, then so does I since I n M+ :::; J 1 n M+. Thus we may assumeI E μ*. Hence Hypothesis 4.2.3 is satisfied. Similarly let I2 := (T n I) (M+ n I).
Then I = hCr(M+/02(M+), so the hypotheses of 4.1.3 are satisfied with I, I2
in the roles of "Io, J 1 ", and hence J 2 E 'T/ by that lemma. Then by construction,
h E μ, so replacing I by I 2 , we may assume I:::; M+T.
Set MH := MnH and LH := (LnH)^00 • As IEμ, 4.2.12.2 says M+ = L ::::l M,
V = [D 1 (Z(R+)),LH] :S LH, LH :S K E C(H) with K i M and K/02(K)
quasisimple, and one of cases (1)-(9) of Theorem C.4.8 holds. We first eliminate
case (9): for in that case, K is the double cover of As with Z(K) = Z(LH); but
then 1 # Z(LH) = Cv(LH) = Cv(L) is LT-invariant, so that K:::; M = !M(LT),
contrary to K i M. Among the :remaining cases, only case (6) is not included
among the conclusions of Theorem 4.2.13-although in cases (5) and (7) of C.4.8,
we still need to show that the extra constraints in conclusions ( 4) and (5) of Theorem
4.2.13 hold. We will eliminate case (6) of C.4.8 later.
In case (5) of C.4.8, LH is a block of type .A.6 with m(V) = 6 and K ~ M24
or He. Therefore for each z E Cv(TnL)#, Vi CK(z), so that conclusion (4) of
Theorem 4.2.13 holds.
Assume that case (7) of C.4.8 holds, so that LH is a G2 (2)-block and K ~ Ru.
We may assume that m(V) = 6, and it remains to show that CK(Vi) i Mn K.
To see this, we will use facts about the 2-locals of K ~ Ru found in chapter J of
Volume I. Observe that Mn K = NK(LH) with (Mn K)/V ~ G2(2). Let V1 be
the (T n LH )-invariant subspace of V of rank 1; then M 1 := CMnK(V1) is of order
3 · 212 , so 3 E 7r(CK(V 1 )) and hence Vi is 2-central in K by J.2.7.4 and J.2.9.1. Let