10.3. THE FINAL CONTRADICTION 755Thus k ~ m-a, so by paragraph two, case (3) of 10.l.l holds with w( G, V) = k = l.
Hence V f:. M9 by E.3.25.
Assume first that Cv(L 0 ) '/= l. Then m(V 1 ) = 4 by l.1.6, so m(Ai) ~ 3 as
k = 1, and hence CA;(Vi) '/= 1 as m 2 (AutM(V 1 )) = 2. But then Vi ::::; M9 by
10.2.9.1, and similarly Vi::::; M9, contradicting V f:. M9.
Therefore Cv(Lo) = 1, so V1 is a TI-set in G by 10.2.10.2. As AutA; (V 1 ) E
A2(AutM(V1), V1), AutA, (V 1 ) is a 4-group of transvections with a fixed axis U 1 , so
Ai~ E4 ~ U1.
Set I:= (~^9 , Vi). We've shown that
Ai =Bi = Nv.9 (V1) '/= 1 '/= U1 = Nv 1 (~^9 ).
'
By I.6.2.2a, 02(I) =Ai x U1 is of rank 4 with C1(Vi) = U1, and as !Vi : U 11 = 2,
J/02(I) is dihedral of order 2d, with d odd. As D2d::::; GL4(2), d = 3 or 5. Now Ai
is of index 2 in ~^9 , so ask= 1, A= AiA2(c) with c = c1c2, where Ci E ~^9 - Ai·
Further as I/02(I) ~ D2d, there is an involution in I interchanging V 1 and ~^9 ,
and U := V n M^9 = U1U2(w), where w = w1w2 with Wr E Vr - Ur· If w acts on
~^9 then 1 '/= [Ai, w] ::::; ~^9 n V, so that V::::; Ca([w, Ai]) ::::; M9 by 10.2.9.1, contrary
to an earlier reduction. Thus w interchanges Li and L~, so by symmetry, Le= L 2.
Now [c, U 1 ] ::::; V^9 is diagonally embedded in V, so we may take z E z# to lie in
[c, U1] ::::; V^9. Then V, V^9 ::::; Gz, so I::::; Gz. Hence as Vr f:. 02(I), V f:. 02(Gz),
contradicting 10.2.11.3. This completes the proof. DCOROLLARY 10.2.13. Case (3) of 10.1.1 holds with w(G, V) = 2 = n(H) for
each H E H* (T, M).
PROOF. Take H E 1i*(T, M). By 10.1.2.3 and 10.2.3.2, Hypothesis E.3.36
holds. By 10.2.9.2, r(G, V) ~ m(Vi) and it is easy to check in each case of 10.l.l
that n'(M) < m(V 1 ). Thus the hypotheses of lemma E.3.39 are satisfied. By
10.2.3.1, n(H) ::::; 2, with n(H) = 1 in case (1) of 10.l.l. Thus by E.3.39.1,
w(G, V) ::::; n(H) ::::; 2, so 10.2.12 completes the proof of the corollary. D
10.3. The final contradiction
LEMMA 10.3.l. (1) Cv(Lo) = l.
(2) Vi is a TI-set in G.PROOF. By 10.2.10.2, (1) implies (2). Thus we may assume Cv(Lo) '/= 1, and
it remains to derive a contradiction. Let H E H* (T, M) and set U := ( Z H) and
H* := H/CH(U). By 10.2.13, case (3) of 10.l.l holds, so by 10.2.8.1, CH(U) ::::;
Ca(Z)::::; M = !M(L 0 T), and hence H* '/= l. By 10.2.13, n(H) = 2, so by 10.2.3.3,
H* ~ 85 wr Z 2 and 02 (H n M)To is a maximal parabolic of Lo. In particular by
10.2.8.1,
[0^2 (H n M), Z]::::; [Lo, Z] = 1.
Then as 3-elements are fixed-point-free on natural modules for L2(4), any I EIrr +(H, U) satisfies either
(a) I= Ii E9 I 2 , where Ii := [I, Ki] is the A5-module for Ki E C(H), or
(b) I= I 1 0 I 2 is the tensor product of A5-modules Ii for Ki.
In either case we compute directly that a(H*, I) = 1. But by 10.2.9, r( G, V) ~
m(V 1 ) and m(V 1 ) = 4 by l.1.6, so s(G, V) = m(M, V) = 2 using B.4.8.2. Set Wo :=
W 0 (T, V). By 10.2.13, w(G, V) > O, so Na(W 0 ) ::::; M by E.3.16. If W 0 = 1, then