15.2. FINISHING THE REDUCTION TO Mr/CMf(V(Mf)) ~ o:t(2) 1117(a) ·u is the sum of isomorphic natural modules, and M 1 is an end-node maximal
parabolic.(b) U is the sum of a natural module and its dual, and M 1 is the parabolic
determined by the interior nodes.
(c) U/Cu(X) is the 6-dimensional orthogonal module for x+ ~ L 4 (2).
(d) U is a 10-dimensional module for x+ ~ L 5 (2).As K:::; Ml, case (b) is eliminated. Let Kr := 031 (M 1 ). Assume case (d) oc-
curs. Then Kr/02(Kr) ~ Z 3 xL3(2). Now X = 031 (I) by A.3.18; so as Ca(Zs):::; I
and Ml = Cx(Zs), Kr= 031 (Ca(Zs)) is T-invariant, and so 031 (0 2 , 3 (Kr)) is a
T-invariant subgroup of 3-rank 1. But this is impossible as T is irreducible on
K/02(K). Assume case (a) occurs. Then as K :::; M 1 , X/0 2 (X) ~ L 5 (2), and
Kr/02(Kr) ~ L4(2). In particular as S acts on M 1 , Sis trivial on the Dynkin
diagram of X/02(X), and so Sis not irreducible on K/0 2 (K). Then by 15.2.21.2,
Kc= 0
31
(Mr), so Kr= Kc. As conclusion (ii) of 15.2.18.3 holds, 031 (Nr) i. Mr,so the minimal parabolic P of X not contained in Kr is contained in Nr. Thus
Y1 := 02 (P) :S Nr with Y1S/02(Y1S) ~ S 3 , but Y1 i. Mr. This contradicts
15.2.21.3.
Thus case (c) holds. In this case, M 1 S =KS is the maximal parabolic deter-mined by the end nodes. We apply an argument made in an earlier reduction, with
Ml, U in the roles of "Mo, Uo", to conclude that for Qr= 02(KS), AutQ 1 (U) con-
tains an FF*-offender. But this is not the case for this parabolic and representation
by B.3.2.6.
This contradiction finally completes the proof of 15.2.23. D
By 15.2.20.3, O(I) = 1, so as F*(I)-=/= 02(I) by 15.2.23, E(I)-=/= 1. By 15.2.20.2,
F*(Mr) = 02(Mri, so there is a component L of.[ with Li. Mr, and by 15.2.20.1,
Lis described in 1.1.5.3. Further as O(I) = 1, Z(L) is a 2-group. Let Lo := (L^8 ),
SL :=Sn L 0 , and ML :=Lon Mr. As usual Lo :::;! I by 1.2.1.3. Recall by our
minimal choice of I that Mr is a maximal subgroup of I; hence I = L 0 Mr. By
1.1.5.3, Z is faithful on L.
LEMMA 15.2.24. L/Z(L) is not of Lie type and characteristic 2.
PROOF. Suppose otherwise. Then we are in one of cases (a)-(c) of 1.1.5.3, andL ~ A 6 in case (c), since Z(L) is a 2-group. ,
Now SL E Syb(Lo) and SL :S ML. Further ML <Lo since Mr is a maximal
subgroup of I= L 0 Mr. So since the maximal S-invariant overgroups of SL in Lo
are parabolics over SL, ML is such a parabolic. Also Z :::; Cr(ML) by 15.2.14.1,
and Z is faithful on L. We conclude from the list in (a)-(c) of 1.1.5.3 that L is
defined over F2, and if L ~ L 3 (2), then ML= SL, so that Ns(L) is nontrival on the
Dynkin diagram of L. Now if m 3 (Lo) = 2 then Lo = 031 (I) by A.3.18 or 1.2.2.a,
so K:::; CL(Z) :::; ML, and hence m3(CL(Z)) ~ 2; but this is not the case for the
groups of 3-rank 2 defined over F2 in Theorem C (A.2.3). Therefore m 3 (L 0 ) = 1,
so Lo= L ~ L 3 (2). But now AutMr(L) is a 2-group, so K centralizes Land hence
m 3 (KL) = 3, contrary to I an SQTK-group. D
We are now in a position to complete the proof of Theorem 15.2.15. By
15.2.20.3, Lis described 1.1.5.3, and indeed appears in one cases (d)-(f) by 15.2.24,