665M, Hypothesis E.6.1 holds by 6.1.3.l applied to Vi in the role of "V". However as
L is transitive on the hyperplanes of Vi, and the stabilizer in LT of a hyperplane
contains a Sylow 2-subgroup of LT, we may take T::; Na(Ui). Thus Ca(f!i) ::; M
by E.6.13, contrary to an earlier observation.This contradiction shows that case (i) holds. The elimination of case (i) will
be lengthier. As L is irreducible on V, V is a TI-set in M, so that by 6.1.3.1,
Hypothesis E.6.1 is satisfied, and we may appeal to results in section E.6.
We first claim r(G, V) > 1. If not, there is a hyperplane U of V with Ca(U) iM, and by E.6.13, U is not T-invariant. Thus U contains the subspace U 0 orthog-
onal to a nonsingular F 4-point of the orthogonal space V. Therefore U contains a2-central involution. As V = R2(LT), V = Sli(Z(Q)), where Q := 02 (LT). Finally
Cv(Nr(U)) ::; U, so as C1'f'(U) = 1,
Sli(Z(Nr(_U))) = Cn 1 (z(Q))(Nr(U)) = Cv(Nr(U))::; U,
contrary to E.6.10.4, establishing the claim that r( G, V) > 1.
Let Mi E M(Ca(Z)), and set Qi := 02(Mi), so that Mi = Nc(Qi). As
H.::; Cc(Z) for HE 7-l(T,M), M-=/:-Mi. Suppose that 02,F(Mi)::; M. Then
Qi= 02(M n Mi)= 02(NM(Qi))
by A.4.4.1, so that Qi E B2(M). By A.4.4.2, C(M, Qi)= Mn Mi, so Hypothesis
C.2.3 is satisfied, with M, Mn Mi, Qi in the roles of "H, MH, R". Now since
V is the orthogonal module and n > 2, L is not a x-block; so for L in the role
of "K", the conclusions of C.2.7 do not hold, and hence L ::; Mn Mi. But thenMi = !M(LT) = M, contradicting Mi -=/:- M. This contradiction shows that
02,F·(Mi) f:_ M.
Next Z ::; R2(LT) = V by B.2.14, so Z = Cv(T). Let X := 051 (NL(TL)).
Then X/02(X) ~ Z5 and XT ::; Ca(Z) ::; Mi, from the structure of 0,4(4) ~
L2(16) and its action on V. Let S := 02(XT), so that S = TLQ. Then Ji(S) ::;
Cs(V) since case (i) holds. Define
1-ls :={Ms ::; Mi : SE Syl2(Ms) and T::; Nd(Ms)}.
As r(G, V) > 1, E.6.26 says Ms::; M for each Ms E 1-ls with n(Ms) = 1.
Now 02(Mi) =Qi::; 02(XT) = S by A.1.6. Then Sis Sylow in 802,F(Mi),so that 802,F(Mi) E 7-ls-and since n(02,F(Mi)) = 1 by E.1.13, 02,F(Mi) ::; M
by the previous paragraph. We saw 02,F* (Mi) i M, so there is Ki E C(Mi)
with Ki i M, and Ki/02(Ki) quasisimple. Let Ko := (K'[) and observe that
X = 02 (X), so X normalizes Ki by 1.2.1.3.
Next as Ki i M, there is Hs E 7-l*(T, M) n K 0 T. Now n(Hs) = 1 by
Hypothesis 6.1.1.2. Thus if S E Syl2(0^2 (Hs)S), then 02 (Hs)S E 1-ls, and hence
Hs ::; M by an earlier remark. Therefore Sis not Sylow in 02 (Hs)S, and hence
S is not Sylow in K 0 S. But if X normalizes T n Ko E Syh(Ko), then T n K 0 ::;
02(XT) = S; thus we conclude X 1 Na(TnKo). In particular, [X, Ki] 102(Ki),
so a Sylow 5-subgroup X5 of X acts faithfully on Ki/02(Ki). Then as X5T = TX5,
this quotient is described in A.3.15. In cases (5)-(7) of A.3.15, X normalizes TnK 0 ,
contrary to an earlier observation. As X/0 2 (X) is of order 5, cases (2) and (4) are
ruled out. So it follows from A.3.15 that either
(a) Ki/02(K1) ~ L2(Pe) and (Mn Ki)/02(Ki) ~ Dpe-€, or