996 14. L 3 (2) IN THE FSU, AND L 2 (2) WHEN .Cr(G, T) IS EMPTYU; hence E has rank 1, and so V = E. On the other hand, U^9 nQ::::; U^9 nG1 = W^9 ,
and Q n W9 = E by 12.8.9.5. Thus (1) holds.
Recall 12 :SJ G 2 =Mand L = 02 (h). As V =Eby (1) and IQI = 2^5 ,
12.8.9.2 says R/V = W/V EB W9 /Vis the sum of m(W/V) = 2 natural modules forI2/R ~ L 2 (2). Therefore R = W9 ~ E 4 and R = [R,LJ::::; L, so that R = 02(L)
as L :SJ I 2. Thus (2) holds. Recall Hypothesis G.10.1 is satisfied; then (3) follows
from part (d) of G.10.1 and the fact that transvections in O(U) have nonsingular
centers.
As R/V is the sum of two natural modules for I2/ R, R has order 26 , and h
has three irreducibles R(i)/V, 1::::; i::::; 3, on R/V. As W/V = CR;v(U), each R(i)contains some ri E W - V. Since U ~ Q~, W ~ Z2 x Ds. Thus we can choose
notation so that (ri) V ~ E 8 for i = 1 and 2, and Z4 x Z2 for i = 3. Then as h
is transitive on (R(i)/V)#, R(l) ~ R(2) ~ E 16 and V = fh(R(3)). It follows that
·A:= R(l) ~ E 16 and A':= R(2) are the maximal element~elian subgroups of
R, AA'= R, and A* is of order 2 in T*, with Cu(A*) =An U a totally singular
line. Thus A* =I= Z(T*), so At= A' fort ET - RQ. Therefore (4) holds as A isIrinvariant by construction, and CH·(A*) ~ Z 2 x 83.
Next [W9, QJ ::::; W9 n Q = V using (1), so 02 (H) = [0^2 (H), W9J centralizes
Q/U, and hence (5) holds. Thus if HA l.s the preimage of CH·(A*), 02 (HA) acts
on AU and hence on A= J(AU), completing the proof of (6). D
From now on, let A be defined as in 14.3.18.4. We will show next that A 6 / E 16 ::::;
NG(A)::::; S5/E32. Set D := Cq(U).
LEMMA 14.3.19. Let K := (0^2 (NH(A)),L). Then
(1) Q = UD.
(2) Either(i) [A, DJ = 1 with Autr(A) ~ Ds, or
(ii) D induces the transvection on A with axis A n U and center Vi.
(3) AutRq(A) E Syl2(AutG(A)), and AutRq(A) ~ Ds or Z2 x Ds.
(4) K is an A 6 -block and A= 02 (K).
(5) CG(K) = 1.
(6) NG(K) = KD, and Dis a subgroup of Ds, with D ~ Ds iff ING(K): Kl=
4 and A< CG(A).
(7) NG(A) = NG(K).
(8) RUE Syb(K).(9) K splits over A.
{10) Aut(K) = K(a,/3), with A(a) = CAut(K)(A) ~ E3 2 the quotient of the
permutation module for K/A modulo the fixed space of K/A, /3 is an involution in-
ducing a transposition on a complement to A in K, and D 8 ~ (a, /3) = C Aut(K) (U).PROOF. By 12.8.4.2, Q centralizes U, while as U ~ Q~, Inn(U) = CAut(u)(U)
by A.1.23, so (1) holds. Next D centralizes the hyperplane An U of A, and [A, DJ ::::;
CA(U) =Vi, so (2) holds.
As A~ E15, AutK(A) ::::; AutG(A) ::::; GL(A) ~ L4(2). As Z =Vi = CA(U)
and RQ E Syl2(NH(A)), Z = CA(RQ), and hence NNa(A)(RQ) :S H, so that
RQ = Nr(A) E Syb(NG(A)). From 14.3.18.6, AutRu(A) ~ Ds and CH(A) is a
2-group, so (2) implies (3), and as Z::::; A, CG(A) = CH(A) is a 2-group.