i062 i4. L 3 (2) IN THE FSU, AND L 2 (2) WHEN .Cr(G, T) IS EMPTY
Pick i to be an H-submodule of UH maximal subject to [UH, K] i. i, and let
UH:= UH/I; then we may take W =[UH, K]. As Li :SJ H, Q* is Sylow in CH·(Li)
by 14.7.5.3, so as Q centralizes V, and UH= (V^0 H•(L~)), UH= [UH,K] = W by
Gaschiitz's Theorem A.l.39. Now by B.4.2, W is either the sum of two natural
modules for K, or the sum of two As-modules for K ~ L2(4). In the first case,
as B centralizes V, V:::::; Cw(B*) = 1, contradicting W = (i/H). _
Thus the second case holds. As K* has no strong FF-modules by B.4.2, I=
CuH(K) by 14.7.30 and F.9.18.6. Then UH= [UH,K] ffiCuH(K) as the As-module
is K-projective, so the lemma holds. D
LEMMA 14.7.43. K* is not U3(8).
PROOF. Assume otherwise. By 14.7.42, Li :::::; K. By Theorems B.5.1 and
B.4.2, H* has no FF-modules, so by 14.7.30 we may apply parts (7) and (4) of
F.9.18 to conclude that UH E Irr+(K,UH)· As q(H*,UH):::::; 2 by Notation 14.7.1,
we conclude from B.4.2 and B.4.5 that UH is the natural module for H*. But then
there is no B-invariant 2-subspace over F 2 satisfying V =[ii, Li]. D
LEMMA 14.7.44. K* is not (S)L 3 (4).
PROOF. Assume otherwise. Again Li:::::; K by 14.7.42.
Suppose first that K ~ SL 3 (4). By 14.7.28, Li= Z(K). Recall Li is inverted
in CTnL(B 0 /0 2 (B 0 )); thus from the structure of Aut(SL 3 (4)), there is t E T
inducing a graph automorphism on K*. Choose I and IH as in F.9.18.4; because
t induces. a graph automorphism, H* has no FF-modules by Theorem B.5.1, so
UH = IH by F.9.18.7, and case (iii) of F.9.18.4 holds. Then as the 1-cohomology
of the natural module is zero by I.l.6.4, UH = i ffi Jt, where i is a natural module
for K and jt is its dual. Further as u~ E Q(H, UH), either u~ is a root group of
K* ofrank 2 with m(UH/CuH(Ua)) = 4, or m(U~);:::: 3 and m(UH/CuH(Ua)) = 6.
If m(U~) = 2 or 3, we get a contradiction from 14.5.18.2, since U~ does not induce
F2-transvections on a subspace of UH of codimension m(U~). If m(U~) = 4 at least
m(UH/DH):::::; 4 by Notation 14.7.1, whereas no subspace of UH of corank at most
4 satisfies the requirement [U~, DH]= A} of F.9.13.6.
Thus K ~ L3(4), and hence H has no module UH with q(H*,UH):::::; 2 by
Theorems BA.2 and B.4.5. This contradiction completes the proof. D
LEMMA 14.7.45. (1) K* ~As.
(2) Either
(a) KE C(Gi), or
(b) Li:::::; Kand K:::::; Ki E C(Gi) with Ki/02(Ki) ~ A7.
PROOF. Conclusion (1) holds if Li i. K by 14.7.42. If Li :::::; K, it holdssince 14.7.43 and 14.7.44 eliminate the other possibilities in 14.7.41.2. Thus (1) is
established.
Next as KE .C(Gi, T), K:::::; Ki E C(Gi) by 1.2.4, so Hi := KiLiT E Hz by
14.7.32. By 14.7.30, Ki/02(Ki) is quasisimple. Applying 14.7.36 to Gi in the role
of "H", Kif 02,z(Ki) is not sporadic. Applying F.9.18.4 to Hi, either Kif 02 ,z(Ki)
is of Lie type in characteristic 2 or Kif0 2 (Ki) ~ A 7. If K =Ki then (2a) holds,
so we may assume K <Ki. Then from the list of possible proper overgroups of As
in A.3.14 with Ki/02(Ki) quasisimple, either Ki/02,z(Ki) is of Lie type over F4
of Lie rank 2, or Ki/02(Ki) ~ A7. In the first case since Kif0 2 (Ki) is defined