i034 i4. L 3 (2) IN THE FSU, AND L 2 (2) WHEN .Cr(G, T) IS EMPTYnow apply F.9.18 to Ki, Gi in the roles of "K,H": As the ot(16)-module in case (i)
of F.9.18.5 does not extend to CJ.i ~ Aut(L 2 (16)) wr Z2, case (iii) of F.9.18.5 holds.
Indeed as Ki~ L 2 (16), subcase (a) of case (iii) holds, so for J E Irr+(Kz,fl,T),
Io:= (IT)= II^8 , and we may choose notation so that i/CJ(Ki) is the natural ororthogonal module for Ki and [i, Kf] = 1.
We claim that Uz = I 0. For if not, case (a) ofF.9.18.6 does not hold and Gi has
no strong FF-modules, so that case (c) of F.9.18.6 does not hold. Thus case (b) of
F.9.18.6. holds, so that Wz := Uz/I 0 and la/CJ a (Kz) are nontrivial FF-module for
Gi, and hence Wz/Cwz(Kz) and io/C1 0 (Kz) are both natural modules for L2(16)
by Theorem B.4.2. Indeed since D 1 < U 1 in case (2) of 14.6.13, we may choose
a as in 14.5.18.5; then Ua E Q( Gi, U), so since Gi has no strong FF-modules by
Theorem B.4.2, u~ is an FF*-offender on both io and Wz. Therefore either Ua is
Sylow in Kz, or interchanging Ki and Kf. if necessary, we may assume that Ua
is Sylow in Ki. In either case, m(U /Cu(Ua)) = 2 m(Ua), so we conclude from
14.5.18.2 that m(U/D) = m(Ua) where D := Da 1 , and that Ua acts faithfully on
fJ as a group of F 2 -transvections with center Za. As Ua is Sylow in Ki or Kz,
and i/C1(Ki) is the natural Ki-module, Ua does not induce a nontrivial group
of F2-transvections on any subspace of io, so fJ n io = C1 0 (Ua) is of codimension
m(Ua) in io, and hence U = IaD. But this is impossible as Ua does not induce a
nontrivial group of F 2 -transvections on Wz. Thus the claim is established.
Set K 2 := Kf.. Since case (iii.a) of F.9.18.5 holds, with J 0 = Uz by the claim,
Uz = Ui + U2 with Ui := [U, Ki], and Uif Cui (Ki) the 2-dimensional natural or
4-dimensional orthogonal module for Kif0 2 (Ki)· Then as UH ~ Uz, we can choosenotation so that 02 (Hi) ~Ki, and hence UH,i ~ ui.
This completes our preliminary treatment of the case K < Kz. In the case
where K = Kz we establish a similar setup: Namely in this case we set Ki :=
02 (Hi) and ui := UH,i·
Thus in any case Z 1 ~ UH,i ~ Ui, so that Z 1 centralizes K 2. Choose g as
in case (2) of 14.6.13, and for X ~ G, let B(X) be the subgroup generated by
the elements of order 5 in X. Then K 2 ~ B(Ca(Z 1 )) = Kf, and by 14.6.13.2,
Z ~ Uf:r,i ~ Uf, so K;, ~ B(CKg(Z)) =Kg. Therefore K2 =Kg, so g E Na(K 2 ).
Set G2 := Na(K2); since g E G-Gi in 14.6.13.2, G2 f:. Gi. Set T2 := NT(K 2 )
and Gi,2 := Gi n G2, so that /Gi : Gi,2/ = /T : T2/ = 2, and in particular
Gi,2 ::::) Gi. As Qi = 02(KzT2), and KzT2 ~ Gi,2, we conclude Qi= 02(Gi,2).
Then as Gi EM by 14.6.1.1, C(G2,Qi) ~ Gi,2 = Na 2 (Qi), so Qi E B2(G2). ThusHypothesis C.2.3 is satisfied with G2, Qi, Gi, 2 in the roles of "H, R, MH". As
Z ~ [U,K2] ~ 02(K2) using 14.6.12.3, F*(G 2 ) = 02 (G 2 ) by 1.1.4.3.
Suppose 02,F• (G2) ~ Gi,2· Then 02(G2) = 02(Gi,2) by A.4.4.1, and we saw
Gi,2 ::::) Gi, so G2 ~ Na(02(Gi,2)) = Gi as Gi EM, contrary to an earlier remark.
Thus 02,F·(G2) f:. Gi,2.
Next Gi = Na(Kz) as Gi E M. If X is an A3-block of G2, then as G2 is an
SQTK-group, /XG^2 J ~ 2; hence Kz = 051 (Kz) normalizes X, and then centralizes
X as Aut(X) ~ 84. Thus X ~ Ca 2 (Kz) ~ Gi,2. Therefore 02,F(G2) ~ Gi,2
by C.2.6, so there is J E C(G2) with J/02(J) quasisimple and J 1:. Gi, 2. If Kz
centralizes Jj0 2 (J), then J normalizes 02 (Kz0 2 (J)) = Kz, contrary to J f:. Gi =
Na(Kz), so we conclude J = [J, Kz]· Furthermore [J, K2] ~ 02 (J) by 1.2.1.2, so as