15.3. THE ELIMINATION OF Mf/CMr(V(Mr)) =Sa wr Z 2 1127as YL :S M2, 02 (M2) = YL is S-invariant. Hence S also acts on L and Ye, so
Ye = 02 (M1). Let t E T - S, and recall [T : Sf = 2 so that t normalizes S. As
Mf = M2, Yi = Ye. Further Ye centralizes L by 0.1.10, and as YL contains a
Sylow 2-group of L, U(L) :S 02(YL)· Then U(L)t :S 02 (YL)t = 02 (Ye) :::1 LS.
Hence (LS,t) = (L,T) acts on U(L)U(L)t, so that (L,T) :::; M = !M(YT) by
15.3.7, contrary to Li. M. D
LEMMA 15.3.19. L/0 2 (L) is not L 2 (2n).
PROOF. Assume otherwise. Then by 15.3.16 and 0.2.7.3, Lis a block of type
L2(2n) or type A5, so by 15.3.18, Lis an A 5 -block. Let Lo := (L^8 ), So :=Sn L 0 ,and Mo := MnLo. Arguing as in the proof of the previous lemma, we conclude that
Mo is the Borel subgroup of Lo containing S 0 , Y+ = YLYe with [YL[ 3 = 3 = [Ye[ 3 ,
Y+/02(Y+) ~ Eg, RL :=Rn LE Syh(L), and WL = U(L) x CwL (L), so
Since U(L) is the A5-module, YL centralizes V by (*),so as Cy(V) =I-1, case (2)
of 15.3.7 holds, with Y/02(Y) ~ 3i+^2. In particular [Oy(V) : 02 (0y(V))[ = 3,
so YL = 02 (0y(V)). Further as Y+/0 2 (Y+) ~ E 9 , Y+ < Y, so that case (2)
of Hypothesis 15.3.10 holds, with Vi = [V, Y+J, and Na(Vi) :::; M by 15.3.11.3.
Now EndL/0 2 (L)(U(L)) ~ F2 so that Ye centralizes U(L). Thus Vi= [V2, Y+] :S
[U(L)CR(L), Ye] :S CR(L). Then L :::; Na(Vi) :::; M, contrary to the choice of
L. D
LEMMA 15.3.20. L/02(L) is not SL 3 (2n) with n > 1 or Sp4(4).
PROOF. Assume otherwise. By 15.3.16, Lis described in 0.2.7.3, and in par-
ticular as W L is an FF-module for L + R+, S is trivial on the Dynkin diagram of L +
by Theorem B.4.2. Further as S normalizes Y, SLY+ = Y+SL, so as each solvable
overgroup of SL in LY+ is 2-closed, Y+ acts on SL. Thus Y+S acts on both maximal
parabolics Pi of L. If Xi := PiY+S i. M, then Xi EH+, contrary to 15.3.19. Thus
L = (P 1 , P 2 ) :::; M, contrary to the choice of L. D
LEMMA 15.3.21. L is not a block of type A6, G2(2), A.6, or A7, and L is not
an exceptional A7-block.
PROOF. Assume Lis one of the blocks appearing in 15.3.21. By 15.3.17, Y+ :S
L, so that Y+ = YL. As Y+SL = SLY+, Lis not of type A 6 or G 2 (2), since. no
proper parabolic in these groups has 3-rank 2. Similarly if L/02(L) ~ A.6, the
preimage of a proper parabolic does not contain 3i+^2 , and if L/0 2 (L) ~ A7, then
L has abelian Sylow 3-groups; thus Y+S / R ~ S3 x S3 in these two cases. Hence L
is not an an exceptional A7-block, since in that case LS/02(LS) ~ A7 rather than
S 7. Further L is not an ordinary A 7 -block, since in that case Mt has no normal
E 9 -subgroup by 0.2.7.3. This leaves the case where L is an A 6 -block, where by
0.2.7.3, s+y: is the stabilizer of a 2-dimensional F4-subspace U of [WL,L]. Now
[WL, L] has the structure of an F 4L-module on which Z(L +) induces scalars in F 4,
and U = U 1 EB U 2 is the sum of two Y+-invariant F 4-points, so Ui = Vi. Thus is
impossible as S interchanges the two F 4 -points in an A 6 -block, but S acts on Vi