i2.9. THE FINAL TREATMENT OF L 0 (2), n = 4, 5, ON THE NATURAL MODULE 86iSuppose first that we are in Case IL Then by the choice made in the first
paragraph of the proof, Ki is one of the groups listed in 12.9.3 other than A 7 ,
L4(2) or L5(2), and H = KiT. Then unless Ki ~ SL2(7)/ E4 9 , Kif0 2 (Ki) isquasisimple, so the hypotheses of F.9.18 are satisfied. But then F.9.18.4 supplies
a contradiction, as none of the groups other than A 7 , L 4 (2), and L 5 (2) appear in
both F.9.18.4 and 12.9.3. Thus we have reduced to Ki ~ SL 2 (7)/ E 49. This case is
impossible, since by F.9.16.3, q(H*,UH)::::; 2, contrary to D.2.17 applied to KiT*
in the role of "G".Thus Case I or III holds, and in either case, H = HiLi with [Li, 02 (Hi)] ::::;
02(Li), so CH(Li/02(Li)) s Hi. As Hi E 1i*(T, M), 3.3.2 says Hi is a minimal
parabolic in the sense of Definition B.6.1. By F.8.5, the parameter b is odd andb;::: 3. Then by F.7.3.2, there is g E (LT,H) = G 0 mapping the edge 1'b-i,1' to
1'o, 1'i, and h E H with 1'2h = 1'0· Set /3 := 1'i9, 8 := 1'h, and let a E {/3, 8}; then
Ua S 02(G 70 m) by F.8.7.2. Therefore as Li :::;! H;::: G 70 m, [Li,Ua] S 02(Li),
and hence Ua S CH(Li/02(Li)) S Hi. Further as Hi is a minimal parabolic,
for each nontrivial Hi-chief factor Ei on U, m(Ua/Cu"' (Ei)) S m(Ei/CE 1 (Ua))
by B.6.9.1. However by 12.8.5.1, each :ff-chief section E on U is the sum of n - 1
chief sections under Hi, so that (n-l)m(Ua/Cua.(E))::::; m(E/CE(Ua)). Hence as
n ;::: 4, if u~ =I- 1 then
2m(U~) < (n - l)m(U~) s m(U /Co(U~)).Now take a = /3. Then Ua = U9 and U = U~, so (*) shows that UH does not
induce transvections on U 7. Therefore by F.9.16.1, D 7 < U 7 , so by F.9.16.4, we
may choose 1' so that u; E Q(H*' fj H). Then taking a = 8' we have a contradictionto (*), completing the proof of (1) and (3), and hence of 12.9.5. D
LEMMA 12.9.6. (1) [Vi, 02(Ki)] =:j:. 1.
(2) h := (02(Gi)^02 ) :::! G2, I2/02(h) ~ 83, 02(h) = C1 2 (Vi), and I2T is aminimal parabolic of LT.
{3) m3(Ca(Vi)) S 1.
PROOF. By 12.9.5.2, Ki/02(Ki) ~ A1, L4(2) or L5(2); and by 12.9.5.1, H :=
KiT E 1iz. Let Q := 02(LT), Qi:= 02(Ki), and H := H/Qi.
Assume that Qi centralizes Vi. Then Qi centralizes (Vl), and by 12.8.8.6,
UH = (Vl), so that Qi centralizes UH. Thus as Ki/02(Ki) is simple, UH E
R 2 (KiQ). Next as Qi centralizes V, Qi SQ< Ri, with Ri/Q the natural module
for Li/Ri ~ Ln-i(2). As H i. M ;::: Na(Q), Qi < Q, so Q =:j:. 1. Therefore
1 =:j:. Q < Ri = 02(Li). As 02(Li) =:j:. 1, Ki is not A1, so that Ki ~ L4(2) or
L 5 (2). As 1 =:j:. Q < 02 (Li), the parabolic LiT of Ki is not irreducible on 02(Li),
so we conclude that n = 4 and Ki ~ L5(2). Then using 12.9.5.3, Ri ~ 2i+B and
Q = 02 (P) ~ E 16 for some end-node maximal parabolic P of Ki- But then
P::::; Na(Q) ::::; M, contradicting Li :::;! Mi. This completes the proof of (1).
Let P 2 be the minimal parabolic of LT nontrivial on Vi, and R := 02(Gi).
Now as Ca(Vi) ::::; Gi and P2 induces GL(Vi),
RG2 = RCa(Vi)P2 = RP2 ~ p 2 ,
so (R^02 ) = h ::::; P 2. Further by (1), R does not centralize Vi, so P 2 = I 2 T and (2)
follows. Finally [h, Ca(V2)]::::; Ch(V2) = 02(V2) by (2), so
2;::: m3(G2) = m3(f2) + m3(Ca(Vi)) = 1 + m3(Ca(V2)),