i3.8. FINISHING THE TREATMENT OF Aa 965u; = UJr :S: VH. Thus as VH is abelian, VH centralizes u;, and hence also u'Y.
Therefore [UH, U'Y] = 1, and hence case (1) of 13.8.8 holds.
Next v; -I 1, so 1 -I v;k = Vjj*. However as V{ = V2, Vjj* ::::1 CH* CV2) = K:J,so V}j/E s::! Vjj* = 02(K:;,) s::! Es, where E := QH n VJ}. But U 0 = VH n V}j :S: E,
so E = Uo as m(Vjj/E) = 3 = m(Vjj/Uo) by parts (3) and (5) of 13.8.29. Also
He :S: Ca(A~) :S: Na(Vjj), so [He, Vjj] :S: Hen Vjj :S: E = U 0 :S: VH, and hence
K = [K, Vjj] centralizes He/VH. Thus to complete the proof of (1) and hence of
(2), it will suffice to establish (3).
Appealing to 13.8.29.5 and the duality in 13. 7.4.2, K has the following noncen-
tral 2-chief factors on QH/He and VH: The natural module UH, its dual QH/He,
and the orthogonal module VH /UH. Therefore Lo has six noncentral 2-chief factors
not in He/VH: two on 02(L 0 ), one each on QH/He and UH, and two on VH/UH.
Therefore by 13.8.30, Lo has at least three noncentral chief factors on He/VH, so
(3) holds and the proof of the lemma is complete. DLEMMA 13.8.32. {1) m(U;) = 1.
(2) m(U'Y n VH) ;::: 3.
{3) Ai :S: VH·
PROOF. By 13.8.31.2, u; -=j:. 1; thus m(UH n Uy) 2: m([UH, U-y]) > 0. Furtherby 13.8.29.5, no member of H* induces a transvection on VH /UH, so
m((U-y n VH)/(U'Y n UH))= m((U-y n VH)UH/UH);::: m([VH/UH, U'Y] 2: 2, (*)
with equality only if (1) holds. In particular this establishes (2), and moving on
to the proof of (1), we may assume that m(U'Y n VH) 2: 4. But then as m(U;) =
m(U-y/(U-y n QH)) :S: m(U-y/(U-y n VH)) and m(U-y) = 5, it follows again that
m(U;) = 1, completing the proof of (1). Thus (1) and (2) are established.
By (1) and 13.8.10, m(UH/DH) = 1, and we have symmetry between 'Yi and
'Yin the sense of Remark F.9.17. By 13.8.31.1, Ai i UH, so by symmetry and
13.8.10.2, Vi i U-y, and u'Y induces transvections on UH with axis DH.
Let /3 E r('Y); by F.7.3.2 there is y E G with 'YiY = 'Y and VY= V,13. By 13.5.4.4,
[Ca(VJ'), VY] :S: Ai, so as Ai i UH,
[CuH(VJ'), VY] :S: UH n Ai= 1. (**)But VJI :S: U-y :S: CH(DH), so V,13 =VY centralizes DH by(**). As this holds for
each /3 E r('Y), v'Y centralizes DH. Therefore v; induces a group of transvections
on UH with axis DH. We saw that no member of H* induces a transvection on
VH/UH, so we conclude from 13.8.18.2 and 13.8.11.l that u; < v;. By parts (1)
and (2) of F.9.13, v; :S: 02 (LiT)x for some x E H. So as LiT is the parabolic
subgroup of H* stabilizing the 2-subspace Vs of the 4-dimensional module UH,
while v; centralizes the hyperplane DH of UH, we conclude that m(V,;) = 2. By
symmetry, EH= VH n Q"k is of corank 2 in VH, so as IUH: DHI = 2, EHUH/UH
is a hyperplane of VH/UH. Thus 1 -=j:. [EHUH/UH,U-y] :S: AiUH/UH by F.9.13.6,
establishing (3). D
We are now in a position to obtain a contradiction, and hence establish Theorem13.8.28. Recall H* s::! L4(2), m(UH) = 5, and L/02(L) s::! A5. Now IQH : (Q n
QH)I = IQHQ : QI :S: I02(LiT) : QI, and as L/02(L) s::! A5, I02(LiT)I = 4.
Next by 13.7.4.2, IQH/He : CqH(Vi)/Hel = IVi/Vil = 4. So as Q n QH :S: