962 i3. MID-SIZE GROUPS OVER Fz
by 13.8.18.4. Also by 13.8.23.1, u; induces transvections on wand UH, so H has
a unique noncentral chief factor on w, and hence w =I. Again by 13.8.23.1, DH
is a hyperplane of UH, so as W =Ii DH, UH =DH and hence Ai = [UH, U 7 ]
and Ai[W, u;J =[UH, u;J is of rank 2. Now u~ = Z(T), so T* acts on [UH, U~]
and centralizes Wi V2 where Wi := [W, U~]. Thus
- -h - * - -
WiAi =[UH, UaJ = Wi Vi,
where h E H with "fh = a. Thus the middle-node minimal parabolic H 0 of H
containing T centralizes [UH, U~], and in particular A~, so H 0 acts on v; since
Gi = H by (2). This is impossible as H 0 ~ 83/D§ has no normal E4-subgroup.
So m(U;) > 1. Now v; ::; 02 (Li), and we've seen that u; is noncyclic and
U~ does not induce a group of transvections with fixed center on i; thus [i, U~]
is the line in i fixed by LJ', and hence [f, U~] = [f, V;]. Therefore by 13.8.9.2
applied to I in the role of "F"' Vi i UT Also we saw v1' centralizes DH n I, so
m(I/DHnI)22.
Suppose m(u;) = m(UH/DH)· Then we have symmetry between 'Yi and"/,
so Ai i UH, and hence [DH, U 7 ] ::; Ai n UH = 1. Further as u; is noncyclic and
we saw earlier that u; does not centralize any hyperplane of UH, m(UH/DH) 2 2.
Hence as m(I/I n DH) 2 2, m(U;) = m(UH/DH) 2 4, contrary to our earlier
observation that m(U;) ::; 3.
Therefore m(U;) > m(UH/DH)· So as m(U;)::; 3, we conclude
3 2 m(U;) > m(UH/DH) 2 2,
where the final inequality holds since we saw m(I/DH n J) 2 2. Thus m(U;) = 3
and m(Us/DH) = 2. Hence UH= DH since m(I/DH n I) 2 2, so [UH,U 7 ] =
[DH, U 7 ] = Ai by F.9.13.6. This is impossible as U~ ::; 02 (Li) with m(Ua) = 3 ..
Thus the proof of 13.8.25 is at last complete. D
LEMMA 13.8.26. If K ~ A 6 , then UH is the natural module for K on which
L1 has two noncentral chief factors or its 5-dimensional cover.
PROOF. In case (1) of 13.8.8, this holds by 13.8.5, so we may assume case (2)
of 13.8.8 holds. Then u; E Q(H, UH), so each noncentral chief factor for K on
UH is of rank 4 by B.4.2 and B.4.5. Suppose K has more than one such factor, and
pick W as in 13.8.22.
First assume u; contains a strong FF-offender on N := UH or W. Then by
B.3.4.2i, U~ = Ri ~ Es is generated by the transvections on N in T. But by
13.8.4.5, VH /UH has a quotient B which is the 4-dimensional H* -module on which
LiT fixes a point. Then as U~ = Ri, U~ is not quadratic on B, contrary to 13.8.4.6.
Thus u; contains no strong FF*-offender on either UH or W, so by 13.8.22,
u; induces transvections on E =: UH or W, and hence m(U;) = 1. This is a
contradiction to 13.8.23.4, as u; = CH·(CE(u;)) for any transvection.
Thus UH has a unique noncentral chief factor. Since UH = (Vl) with V 3 =
[i/3,L1] a nontrivial irreducible for Li, UH/CoH(K) is the 4-dimensional natural
module on which Li has two noncentral chief factors. Then by I.1.6.1, UH is either
natural or a 5-dimensional cover, completing the proof. D
LEMMA 13.8.27. (1) Either