i3.8. FINISHING THE TREATMENT OF A 6 963(a) L/02(L) 9:! A5, H 9:! A6 or 86! and UH is the natural module for K
on which Li has two noncentral chief factors or its 5-dimensional cover, or.
(b) L / 02 ( L) 9:! A5, H* 9:! L4 ( 2), and UH is a 4-dimensional natural module
for H*.
(2) Gi = H = KT.
(3) If case (1) of 13.8.8 holds then DH= UH, D-y = U-y, V induces a group of
transvections on U-y with center Vi, and Vi S UT Further V-y i QH, so we have
symmetry between ry and 'Yi.
PROOF. By 13.8.24 and 13.8.25, the list of 13.8.21.2 has been reduced toK* 9:! A6, L4(2), or G2(2)'. Further K s Ki E C(G1) by 1.2.4, and as Gi E Hz,
Ki/02(Ki) 9:! A5, L4(2), or G2(2)'. So as A.3.12 contains no inclusions between
any pair on this list, we conclude that K =Ki. Thus Gi = KiT =KT= H by
13.8.21.3, so (2) holds.
By (2) and 13.8.7, DH = UH and D-y = U-y. Thus V induces a group oftransvections on U-y with center Vi by F.9.16.1, so Vi S UT Thus to complete the
proof of (3), we assume V-y s QH and derive a contradiction. Then [UH, V-y] S Vin
Ai= 1. Thus V^9 S Ga(V3) s Mv, so that [V, Vf] =Vi= [V, V^9 ]. Then Gvg(V)
is a hyperplane of V^9 and hence conjugate to Vf, so VS Ga(Gvg(V))::::; Mf, by13.5.4.4. Then Vi = [V, V9] s V n V^9 , contrary to 13.8.3.
It remains to prove (1). However if K* 9:! L4(2) or A 6 , then (1) holds by
13.8.25 or 13.8.26, so we may assume that K* 9:! G 2 (2)' and derive a contradiction.
Thus case (2) of 13.8.8 holds as G 2 (2)' does not appear in 13.8.5. As H* has no
strong FF-modules and no transvection modules by B.4.2, 13.8.22 and 13.8.21.4 say
UH E Irr +(K, UH)· So as u; E Q(H*, UH) by 13.8.8, B.4.2, B.4.5, and I.1.6.5 saythat UH is the 7-dimensional Weyl module or its 6-dimensional quotient module.
Thus m(UH) S 8.
By 13.8.18.2 and 13.8.11.1, u; < v;. Thus m(U;) < m2(H*) = 3, so by
B.4.6.13, ru• 'Y' u-H > 1. But by the choice of ry in case (2) of 13.8.8, m(U:;) I ;::::m(UH/DH), and [U-y,DH] S Ai by F.9.13.6, so we conclude Ai S UH. Thus
HJ_ := CH (Ai) = GH(Ai) is a maximal parabolic of H, and u; is elementary
abelian and normal in GH(Ai). Therefore as m(U;) < 3, u; 9:! E4 (cf. B.4.6.3).
Thus m(D-y n UH) 2 m([U-y, UH]) 2 3. Next by 13. 7.4.2, QH J Ho is H* -isomorphic
to UH/GuH(QH), so 1 # [GqH(Ai)/Ho,U-y]·s D-yHo/Ho, so D-y i Ho. Finally
as VH is abelian, VH s Ho, and by (2) and 13.8.4.5, H* 9:! G 2 (2)' or G 2 (2) is
faithful on VH/UH; so as u; 9:! E4, m((D-y n VH)UH/UH) 2'm([VH/UH, U-y]) 2 3.
Thus as ru• "I' u H > 1,
m(U-y) > m(u;) + m((D-y n VH)UH/UH) + m(D-y n UH) 2 2 + 3 + 3 = 8,
contrary to the previous paragraph.
THEOREM 13.8.28. K* 9:! A5.
DUntil the proof of Theorem 13.8.28 is complete, assume G is a counterexample.Then H* 9:! L4(2), L/02(L) 9:! A5, and m(UH) = 5 by 13.8.27.1. Recall G 2 =
2.Na(Vi), and set K2 := 0 (NH(V2)), Q2 := 02(G2), and G2 := G2/Q2. Set
Uo := (Ujf^2 ) and Vo := (Vf/^2 ).
Since L/02(L) 9:! A5, by 13.5.4, I2 = 02(Gi)L2,+ ::;! G2 with 02(I2) =
Gr 2 (V2) = h n Q2 and j2 9:! 83. Let g E L2,+ - H, so that V{ = V 2.