948 i3. MID-SIZE GROUPS OVER F2If 02 (K) = L 0 , then H ::::; Na(0^2 (K)) = N 0 (L 0 ) ::::; M by 13.2.2.9, contrary to
Hf:. M. If 02 (K) =Li or Li,+, then UK = %, so H :'S: Gin G3 :'S: M by 13.5.5
for the same contradiction. Thus [K*, Li] = 1, so% = [V3,Li] :'S: [UH,Li] :'S: CuH(K),
and then as K :'.::] H, UH= (Vl)::::; CuH(K), contrary to K* f. l. D
LEMMA 13.8.6. Assume H =KT with K E C(H), K* ~ A5, and UH is a
natural module for K* or its 5-dimensional cover. Let K2 := 02 (CH(Vi)) and
U2 := (VK^2 ). Then
(1) V 2 is generated by a vector of weight 4 in UH and K2T/02(K2T) ~ 83.
(2) [K2,L2]::::; 02(K2) n 02(L2).
(3) U2 = (U2, L2] :'S: UL.
(4) If m(UH) = 4 and LT~ 85, then m(U2) = 6 and UL/V has a quotientisomorphic to the 16-dimensional Steinberg module for LT.
(5) If m(UH) = 5 and Ui := CuH(K), then m(U2) = 8, Uo := (Uf) :'S: UL, and
U 0 /V is a quotient of the IS-dimensional permutation module for LT on LT /LiT.
(6) L/0 2 (L) ~ A5.PROOF. Observe that (1) and (6) hold by 13.7.6.3. In particular, K2,T* is the
parabolic of H* stabilizing the point i/2 generated by a vector of weight 4, and V3
is a line with all vectors of weight 4.
By (6) and parts (1) and.(6) of 13.5.4, L2 :'.::] G2, so [L2,K2]::::; CL 2 (V2) =
02(L2), and hence (2) holds. Now
U2 = (VK2) = (VaL2K2) = (VaK2L2)::::; (U.f?)::::; UL,and as L2 :'.::] L2K2 and V = [V, L2], U2 = [U2, L2], so (3) holds. Set U2 := U2/Vi;
it follows that m(U 2 ) = 2m( (V{^2 ) ). Thus m(U 2 ) is 6 in case (4), and 8 in case (5).
Assume the hypotheses of (4), and recall V <UL by 13.8.4.7. Let V::::; W <UL
with LT irreducible on UL/W. By 13.8.5.2a, UH= [UH, Li], so that UH= [UH, Li]
since Vi= [Vs, 02(Li)]; hence UHV/V = [UHV/V, Li]~ E4. As W <UL= (Ufi),
UH i. W, so that UHW/W is LiT-isomorphic to UHV/V. Similarly by (3), U2 =
[U2, L2] ::::; UL, and as we sayv-m(U 2 ) = 6 in this case, U2fV = [U2/V, L2] ~ E4, soU2 W/W is L2T-isomorphic to U 2 /V. Hence. ( 4) holds by G.5.2.
Finally [Ui, LiT] ::::; Vi ::::; V, so (5) holds. D
LEMMA 13.8.7. Assume H = Gi. Then DH< UH iff D'"Y < U'"Y.
PROOF. Assume the lemma fails. If DH =UH but D'"Y < u'"Y, then u'"Y i. QH,
and in particular V'"Y f::. Q H. Thus there is some ,8 E I' ( ')') with V,e f::. Q H. By F. 7. 9 .1,
d(,B, ')'i) = b. Thus we have symmetry (cf. the first part of Remark F.9.17) betweenthe edges ')'o, 'Yi and ,B, ')', so we may assume that DH < UH but D'"Y = UT Then
case (i) of F.9.16.l holds, so that UH induces a nontrivial group of transvections
on U'"Y with center Vi. Recall there is g E Go:= (LT,H) with ')'g ='Yi, and setting
a:= 1'i9 and Ua = UfI, U~ -:/= 1 but [UH, Ua] = V{ =:Ai. Then Ua induces a group
of transvections on UH with center Ai, so by 13.8.5, H =KT for some KE C(H),