13.8. FINISHING THE TREATMENT OF Aa 955As n is odd, B =Be, so B centralizes V 3. Therefore as C1H (B) = 0, we conclude
V3 i. IH. If IH = [UH, K], then ViIH is invariant under K L 1 T = H, so UH = ViIH.Then as To centralizes V3, UH= lH ffi CuH (K) and CuH (K) = CuH (B) = V 3. But
now UH= ('VsH) = V3, contrary to 13.5.9. Hence K* is faithful on UH/IH, so case
(b) or ( c) of F. 9 .18.6 holds with l H in the role of "W". Therefore [UH, K] /I H is anFF-module for KT, and hence this quotient is also the tensor product of natural
modules for K* and 83. Then again B centralizes V 3 , but is fixed-point-free on
[UH,K], so that Vi i. [UH,K]. Now we obtain a contradiction as in the earlier
case, arguing on [UH, K] in place of I H.Therefore (b) holds. As q(H,UH)::::; 2, K is not L 3 (4) by B.4.5. Thus
K* ~ L2(4), 8L3(4), or (8)U3(8). We claim L 1 ::::; K; so assume otherwise. As
1 =/:- 0
31
(B) ::::; L1 but L1 i. K, it follows that L/02(L) ~ A.6 and IBl3 = 3.Hence K* ~ U3(8) or L2(4). In the first case, A.3.18 supplies a contradiction
as L1/02(L1) ~ E 9 and T acts on L 1 but does not permute with the subgroup
generated by the element x in A.3.18.b. Thus K ~ L 2 (4), and as Lo and L 1 ,+
are the only proper T-invariant subgroups of L 1 which are not 2-groups, K Li =
K x L 0 , where Le = Lo or Li,+· In the former case, K ::::; Na(Lo) = M by
13.2.2.9, a contradiction. In the latter case, as [Lo, t] ::::; 02 (L 0 ) and case (a) fails,
we have a contradiction. Thus the claim is established.
By the claim and 13.7.5.2, UH = [UH, K]. We next ·observe that K* is not
(8)U 3 (8): For otherwise we may apply F.9.18.7 and B.4.5 to conclude that UH
is the natural module for K* ~ 8U3(8), defined over Fs. But then there is no
B-invariant subspace V3 = [if3, L1] of 2-rank 2.
Suppose K* is 8L3(4). By B.4.5, any IE Irr +(K, UH, T) is the natural mod-
ule. Further B = B(B) ::::; L1 by 13.7.3.9, and Bis of 3-rank 2, so L/02(L) ~ A 6.
Then as X = Li,+ is inverted by t E Cr(Lo/0 2 (Lo)), we conclude that either t
induces a graph-field automorphism on K with L 0 = CLi(t) = Z(K*), or tin-
duces a graph automorphism on K and X = [Li, t] = Z(K). In the first case,
H::::; Na(Lo) ::::; M by 13.2.2.9, contrary to Hi. M; so the second case holds. Now
case (iii) ofF.9.18.4 holds, with JH = itIJft, where J E Irr+(UH,K, T) is a natural
module for K* and Jt is its dual. By F.9.18.7, JH = [UH, K], so IH =UH by the
previous paragraph. Further u'Y E Q(H, UH) is either a root group of K* of rank
2 with m(UH/CoH(U 7 )) = 4, or m(U;) ~ 3 with m(UH/CoH(U 7 )) = 6. In the
first case by F.9.16.2, u; is faithful on DH of corank 2 in UH; and in the second,
at least m(UH/DH) ::::; m(U;) ::::; m 2 (H*) = 4. In either case, no subspace DH of
this corank in UH satisfies the requirement [u;,.DH] = A1 of F.9.13.6.
We are left with the case Li ::::; K* ~ L2(4). Thus L/02(L) ~ A6 by 13.7.5.5.
As L 1 = [L 1 , T] and H = KL1T = KT, H* ~ 85. Then as case (2) of 13.8.8
holds, VC: ::::; Ri E 8ylz(K*), and m2(Ri) = 2, so VC: = Ri by 13.8.18.4. Now by
13.8.4.5, VH /UH is a nontrivial quotient of the 5-dimensional permutation module
for H* ~ 85. Then as v; = Ri, v; is not quadratic on VH/UH, contrary to
13.8.4.6. D
LEMMA 13.8.21. {1) L1 ::::; K.
(2) K/02(K) is Ln(2), 3::::; n::::; 5, A6, A1, or G2(2)'.
{3) H =KT. In particular if K ~ L3(2), then H* ~ Aut(L 3 (2)).