1072 i4. L 3 (2) IN THE FSU, AND L 2 (2) WHEN .Cr(G, T) IS EMPTYor Sp 4 (4), or else Lu/0 2 (Lu) ~ SL 3 (4). The first case is impossible as K has two
noncenfral 2-chief factors. In the remaining two cases, there is Y of order 3 in
CLu(K/0 2 (K)), so Y:::::; Nc(K) = H, a contradiction as CH(K/02(K)) = QH by
Theorem 14.'7.63. D
LEMMA 14.7.69. U~ is of order 2.
PROOF. Assume otherwise. Then as U~ :::::; Q ~ E4 by 14.7.66, U~ = Q.
Therefore using Remark 14.7.64,
(a)so as
(b)
we conclude
(c)From the action of H* on U, for u EU - Uo V we have m([u, Ua]) ~ 2, so [u, Ua] 1:.
Za. Thus we conclude from 14.7.4.1 and (c).that
UnQa=UoV=Cu(Ua)· (d)
Then m(U~) = 2 = m(U /Uo V) = m(U /UnQa), so that we have symmetry between
'Yi and a as discussed in Remark 14.7.39. As U:::::; Ga= Cc(Za) and Cc(U):::::; QH:
Za:::::; QH n Ua. (e)
Suppose first that Vi:::::; Ua. Then Vi:::::; UnUa = F, so F = UoV by (c). Hence
m(U~) = 2 = m(U/F) = m(Ua/F), so
QHnUa=F:::;U.
Then using (e), Za:::::; U. It now follows from 14.7.4.1 that m(QH/CQH(Za)):::::; 1.
But CQH(Za):::::; Nc(Ua) since H = Cc(z), so [CQH(Za), Ua]:::::; QHnUa:::::; U. This
is impossible, since by 14.5.21.1, QH /CH(U) is dual to U/Cu(QH) as an H-module,
so Ua centralizes no hyperplane of QH /CH(U).
Therefore Vi 1:. Ua. Hence Vi 1:. F, so we can now refine (b)-(d) to:
[U, Ua] =Un Ua = F and F x Vi= UoV = Cu(Ua) =Un Q°'. (f)
Suppose that Uo = Vi. Then by (f), Fis a hyperplane of V = Cu(Ua), and
by symmetry between 'Yi and a, Fis a hyperplane of Cu,,(U) and Cu,,(U) E vc.
Hence by 14.7.67.2, Cu(Ua) = V = CuJU), so that Vi :::::; Ua, contrary to our
assumption.
Therefore Uo > Vi. By I.1.6.2, m(U 0 ) :::::; 2, so that m(U 0 ) = 2 or 3.
Suppose first that m(Uo) = 3. Then U is the module U discussed in Remark
- 7.64. In particular the 2-dimensional F 4 -subspace F = C~) is partitioned^4
by V, Uo, and the three I-dimensional F 4 -spaces spanned by the various [ii, s*] for
s* E Uff and ii EU - U 0 V. So as Cu(Ua) = F x Vi by (f), F has the partition
F = F 0 U Fi U Fv,
where Fv := F n V, Fo := F n Uo, and
Fi:= {[x,y]: x E Ua -QH,Y EU - UoV}.
(^4) Following Suzuki, a partition of a vector space is a collection of subspaces such that each
nonzero element is contained in a unique subspace.