ia.s. FINISHING THE TREATMENT OF A 6 967that IVH: VH n Hll::; 2, and [Uk, VH n H^1 ]::; Uk n QH =Uk n V with VUH/UH
of rank 1 by 13.8.4.5. Thus 02 (Li) induces transvections with a common center
on (V H n H^1 ) UH/UH of index at most 2 in V H /UH. So we conclude that K has atmost one nontrivial chief factor on VH/UH, and such a factor must be the natural
module on which Li has one noncentral chief factor. So since Li has two noncentral
chief factors on UH, and QH/Hc is H-isomorphic to UH of rank 4 by 13.7.4.2, we
conclude that Li has at most six noncentral 2-chief factors. However by 13.8.6.4, Li
has at least six noncentral chief factors on UL/V, and hence at least eight noncen-
tral 2-chief factors including those on 02(Li) and V. This contradiction establishes
the claim that UL is abelian.
Since UL is abelian, UL ::; He. Also we saw Ai ::; UH, so UL ::; Ca(Ai) =
H^9 ::; Na(U-y) as H = Gi. But also U-y ::; M, so UL and U-y act quadratically on
-1 -1 - -1 -1each other. In particular, Uf ::; Q}r ::; 02(CH•(Vi9 )), so m(Uf *)::; 2, as
- -1 -. -1
02 (CH• (Vt ) ) ~ Es is not quadratic on UH. Indeed as V i Q}r, 1 =f V9 * ::;
-1
Uf *,so IUL: V(ULnQ}r)I::; 2. Thus as [ULnQ}r, V:f]::; Ai, there is a subgroup
B /V of index at most 2 in UL /V such that [V:f, B /VJ ::; Ai V /V. In particular,
CuL;v(V:f) is of codimension at most 2 in UL/V, so as m(H*, S) = 8 for the
Steinberg module B for H*, we conclude from 13.8.6.4 that m(U H) = 5.
Define Ui and Uo as in 13.8.6.5, and recall that V ::; Ua. Assume first that
Uo <UL. Then as Li is irreducible on UH/VaUi, VaUi =UH n Uo, so the image of
UH in UL/Uo is a T-invariant 4-group. Similarly define U 2 and K 2 as in 13.8.6.5,
and set U2,i := (V 3 K^2 ). Then U2,i = Vf in the 5-dimensional orthogonal space
UH, so U2,i/V2 ~ Es with IU2,i : Ui Val = 2 and UH = (Uf,i). By 13.8.6.3,
U2 = [U2, £2], and by 13.8.6.5, m(U2) = 8, so U2/Vz = U2,i/Vz Ee UJ,i/Vz for
l E £2 - H and UiUfV ::; Uo n U2 with £2 irreducible on U2/UiUfV ~ E4. As
UH= (Ufi) and UL> Uo, U2 i Uo; so Uo n U2 = UrUfV, and hence Uz/(U2 n Uo)
is also a T-invariant 4-group. We conclude just as in 13.8.6.4 that UL/Uo has a
Steinberg module as a quotient, and then obtain a contradiction as in the previous
paragraph.
Therefore Uo =UL. As UH= [UH, Li], UL= [UL,L]. By 13.8.6.5, UL/V =
U 0 /V is a quotient of the 15-dimensional permutation module on L/ LiT; so as
UL/V = [UL/V,L], G.5.3.3 says that either UL/Vis £-isomorphic to V, or UL/V
has a quotient UL/ E isomorphic to the 5-dimensional cover of V. Indeed as V:f
centralizes a subspace of UL/V of codimension at most 2, in the latter case G.5.3
implies that V = E.
So m(UL/V) = 4 or 5, and hence m(UL) = 8 or 9. If m(UL) = 8, then UL= U2
by 13.8.6.5, so K2 ::; Na(UL) ::; M = !M(LT), and then H = (K2, LiT) ::; M,
contrary to HE 1-lz. Therefore m(UL/V) = 5.
Let ui E Ui - V. Suppose Ui ::; Z(Q). Then UL = Uo ::; Z(Q). Also
Wi := (uf) is a quotient of the 15-dimensional permutation module on L/ Li(TnL)
with Wif(Wi n V) ~ UL/V of rank 5, so we conclude from G.5.3 that Wi is the
5-dimensional cover of a copy of V. This is contrary to Theorem 13.4.1 and our
choice of Gas a counterexample to Theorem 13.8.1.
Thus Ui i Z(Q), so that IQ: CQ(ui)I = 2. Now UL is generated by Vanda
set I of 5 conjugates of ui, so
CQ(UL) = n CQ(i).
iEI