1064 14. L 3 (2) IN THE FSU, AND L2(2) WHEN .Cf(G, T) IS EMPTY
14.7.40, so the lemma holds. On the other hand, if K/02,z(K) is defined over
F 2 , then from F.9.18.4, either (2) holds, or K/0 2 (K) ~ A 6 , and T is trivial on
the Dynkin diagram of K/0 2 ,z(K) from the possible modules listed in that result.
However in the latter case, Li= Z(K*) by 14.7.28, so as Tis trivial on the Dynkin
diagram of K/0 2 ,z(K), KT is generated by solvable overgroups of LiT, which lie
in M by Theorem 14.7.29, contrary to H i. M. Thus (2) is established, and it
remains to establish (1) when K/02(K) is not A5.
If m 3 (K) > 1, then K = 0
31
(H) by A.3.18, so (1) holds. Thus we may as-
sume m3(K) = 1, so as K is not A5, K ~ L3(2) by (2). Assume Li i. K.
Then Li centralizes K* as Out(L 3 (2)) is of order 2 and L1 = [L1, T]. Now if
H /CH (K*) ~ Aut(L 3 (2)), then His generated by a pair of solvable subgroups
containing LiT which lie in M by Theorem 14.7.29, contrary to H i. M. There-
fore H /CH·(K) ~ Aut(L 3 (2)), so KT has no FF-modules by Theorem B.4.2.
Therefore by parts (7) and (4) of F.9.18, either fJH E Irr +(K, UH) or UH= i +ft
with i a natural K -module and t inducing an outer automorphism of K. In either
case, CGL(UH)(K) = 1, impossible as Li centralizes K. D
LEMMA 14.7.48. (1) There is a unique KE C(H), and H =KT.
(2) UH= [UH,K].
PROOF. By Theorem 14.7.29, H is not solvable, so there exists K E C(H).
By 14.7.47.1, L 1 is contained in each KE C(H), so K is unique. Then CH (K)
is solvable by 1.2.1.1, and hence CH (K) = 1 by 14.7.28, since Li S K but
Li i. Z(K) by 14.7.47.2. So (1) holds as Out(K) is a 2-group in each case listed
in 14.7.47.2.
As Li S K, V = [V,L1] S [UH,K], so UH
holds.
LEMMA 14.7.49. K* is not L 3 (2) or A 6 •
(VH) = [UH,K], and (2)
DPROOF. Assume otherwise. First H =KT by 14.7.48.1. By 14.7.11, H is not
L3(2), A5, or 85. Thus Tis nontrivial on the Dynkin diagram of K, a contradiction
as H = KT and T acts on L 1. D
LEMMA 14.7.50. K* is not A 7.
PROOF. Let i be a maximal submodule of UH, and UH := UH/i. As UH =
[UH' K] by 14. 7.48.2, u H is a nontrivial irreducible for K. As u~ E Q(H*' u H) by
Notation 14.7.1, UH is of rank 4 or 6 by Theorems B.4.2 and B.4.5.
We first eliminate the case dim(UH) = 4. Notice H* ~ A1 since UH is not
invariant under 87. By 14.7.2.1, Vis isomorphic to Vas an L 1 T-module, so from
the action of H on UH, NH (V) is the stabilizer H4 3 in H* ~ A 7 of a partition
of type 4, 3. Set HM := H n M; by 14.3.3.6, HM = 'NH(V). As H4 3 is solvable
and maximal in H*, we conclude from Theorem 14.7.29 that HM=' H4, 3 • Since
M = LCM(L/02(L)), an element t E T n L inverts Li/0 2 (L1) and centralizes02 (CHM(L/0 2 (L))) modulo 02 (M). This is a contradiction as H* ~ A1 rather
than 81, so elements of H4, 3 - 02 (H4, 3 ) invert 02 (H4, 3 )/0^2 (02(H4, 3 )).