1048 i4. L 3 (2) IN THE FSU, AND L2(2) WHEN .Cf(G, T) IS EMPTY
REMARK 14. 7 .17. Notice ( 6) shows that our hypotheses are symmetric between
'/'i and )', in the sense discussed in Remark F.9.17; therefore if a result 8('/'i, 1')
(proved under the choice m(U;) 2:: m(UH/D~) > 0 made in Notation 14.7.1)
holds, then 8(1', 'l'i) also holds. Similarly as a is an H-translate of)', 8('/'i, a) and
8(a,'/'i) hold too.PROOF. (of 14.7.16) By 14.7.15.1, we may apply 14.7.5 to H, so 14.7.5.5 says
that U = [U, Li].
From Notation 14.7.1, Ua ::::; Ri, so as U is elementary abelian, (2) follows from
14.7.15.2. Then from our choice of)',
1 = m(U;) 2:: m(U/D) > 0,
and hence m(U/D) = 1. Thus we have established (6), and hence also the sym-
metry between '/'i and 1' discussed in Remark 14.7.17. As Z2 ~ U~ E Q(H*,U),m(U/Co(Ua))::::; 2. Then as U = [U,L], (1) holds by D.2.17.
From 14.7.15.2 and the action of Aut(K*) on the module U for K* in (1),
[U, Ua] = Co(Ri) is of rank 2; then as Vis of rank 2 and centralizes Ri = Q*, we
conclude that [U, Ua] = V = c 0 (Q*). Therefore u; does not induce transvections
on U, so u; does not centralize D, and hence Ai ::::; [U, Uy] by F.9.13.6. Thus bysymmetry between '/'i and a, Vi ::::; [U, Ua], so that [U, Ua] = V, completing the
proof of (3). In particular Aq ::::; V, so as H = Gi, (5) follows from 14.7.3.4, and
(4) follows from (5) and 14.7.12.2.By (1) and 14.5.21.1, Li has at least six noncentral 2-chief factors, so (7) follows
from 14.7.10. DLet E := [U, Q] and R := (EL). By 14.7.16.1, Uhas the structure of a 3-
dimensional F 4 -module preserved by KQ, with the 1-dimensional F 4-subspaces
the Li-irreducibles since Li= Z(K*). Thus Vis a 1-dimensional F4-subspace, and
from the action of Q* on U, E = Co(U~) is a 2-dimensional F4-subspace. Then
as Vi ::::; [U, Ua] by 14.7.16.3, m(E) = 5. Set EH := Eh-
1
, so that EH = Co(U;).
Define E'Y by E'Y/Ai = Cu 7 /Ai (U), and set Da := D~ and Ea := E~. Observethat these definitions of "EH, E'Y" differ from those in section F.9, but the latter
notation is unnecessary here, since U, D play the role of the groups "VH, EH" of
section F.9.LEMMA 14.7.18. EH= Cu(U 7 ) is of index2 inD, E'Y = Cu 7 (U), E = Cu(Ua),
and Ea = Cu°' (U) is of rank 5.
PROOF. ByF.9.13.7, [D,D'Y] = 1, whilem(U/D) = 1 = m(U'Y/D'Y) by 14.7.16.6.
Also for x E U'Y -D'Y, [x, DJ :S Ai by F.9.13.6, so m(U/Cu(U'Y)) :S m(D/CD(x)) +
1::::; 2. Thus as Cu(U'Y)::::; EH and m(U/EH) = m(U/E) = 2, these inequalities areequalities, and so the first statement of the lemma follows. Then the second state-
ment follows from the symmetry between '/'i and 1' in Remark 14.7.17, and then
the third and fourth statements follow from the first and second via conjugation by
h. DLEMMA 14.7.19. (1) [S,Q] = R.(2) [R,Q] = V.
(3) R::::; Z(S); in particular, R is abelian and R::::; CH(U).
(4) <l?(S) = [S, SJ = V.