i4.7. FINISHING L 3 (2) WITH (VGl) ABELIAN 1065containing a subgroup isomorphic to A6 or L3(2). Thus either HivI = H4, 3 , or HivI
is the stabilizer H;s,i of a partition of type 23 , l.
Assume first that HivI = H4, 3 , so that H* ~ 87. As V = [V, Li] is a T-invariant
line, Li ~ Z3 fixes 4 points, and V2 = (e5,5). Set Y := 02 (H 2 s,i); then (V{) is of
rank 3, contrary to 14.7.2.2.
Finally assume that HivI = H;s,i· This time as Vis a line, f'2 = (e1,2,3,4),so that [f'2, 02 (H4,3)] = 1, and then [V 2 , 02 (H 4 , 3 )] = 1 by 14.7.2.3. But then
m3(Cc(Vz)) > 1, contrary to 14.7.4.3. D
LEMMA 14.7.51. K* is not Ln(2).
PROOF. Assume otherwise. In view of 14.7.49 and Theorem C (A.2.3), n = 4
or 5. Observe P := Li(T n K) is a T-invariant minimal parabolic of K.
Assume first that T is nontrivial on the Dynkin diagram of K. Then n = 4,
P is the middle-node parabolic, and H ~ 88 • Define i and UH as in the proof
of 14.7.50. Again using Theorems B.4.2 and B.4.5, we conclude that m(UH) = 4
or 6, and since P* acts on the T-invariant line V, that UH is the 6-dimensional
orthogonal module for H*, and Vis a totally singular line. Thus m 3 (CH* (V 2 )) = 2,
so m3(CH(Vz)) = 2 by 14.7.2.3, again contrary to 14.7.4.3.Thus T is trivial on the Dynkin diagram of K, so K = H. Thus H
is generated by rank-2 parabolics Hi containing P* which satisfy Hi/0 2 (Hi) ~
L 3 (2) or 83 x 83. Therefore Hi :::; M by 14.7.49 or Theorem 14.7.29, contrary to
H~M. D
THEOREM 14.7.52. (1) H = KT = Gi is the unique member of 1-lz, and
UH=U.(2) K* ~ A5 or G2(2)'.
PROOF. Part (2) follows since 14.7.49-14.7.51 eliminate all other possibilities
from 14.7.47.2. As K E C(Gi,T), K :::; Ki E C(Gi,T) by 1.2.4. But Gi E
1-lz, so since Ki/02(Ki) is quasisimple by 14.7.30, (2) shows there is no proper
containment K < Ki in A.3.12, and hence K =Ki E C(Gi). Then by 14.7.48.1
applied to both Hand Gi, H =KT= Gi, so (1) holds. D
14.7.4. Eliminating the case 02 (H*) isomorphic to G 2 (2)'. In the re-
mainder of this section, set Mi := H n M. Thus Mi = CM(z) as H = Gi by
Theorem 14.7.52. Further Mi = NH(V) by 14.3.3.6. Abbreviate UH by U. Since
in this subsection we use o: in preference to "(, we will reserve the abbreviation D
not for DH= Un Q'Y but instead for Un Qa.
In this subsection we show K* ~ A5 by proving:
THEOREM 14.7.53. Kj0 2 (K) is not G2(2)'.
Until the proof of Theorem 14.7.53 is complete, assume His a counterexample.
Recall we are operating under Notation 14.7.1, so we choose"( as in 14.5.18.4 and
0: as in 14.5.18.5, and in particular u; E Q(H*, U).
LEMMA 14.7.54. U is either the 7-dimensional indecomposable Weyl module
for K* ~ G2(2)', or its 6-dimensional irreducible quotient.
PROOF. By 14.7.48.2, U = [U, K]. By Theorems B.4.2 and B.4.5, the 6-