968 13. MID-SIZE GROUPS OVER F2Therefore as IQ : Cq(ui)I = 2, m(Q/Cq(UL)) ::::; 5. Also CL(ui V/V) = LiT, so
CLr(ui) is of index 2 in LiT. We conclude from G.5.3.3 that Q/Cq(UL) is a copy
of V as a L-module, or its 5-dimensional cover. Therefore Li has one noncentral
2-chief factor on each of 02 (1 1 ), Q/Cq(UL), UL/V, and V. Let k and j be the
number ofnoncentral chief factors of Li on Cq(UL)/UL and Ho/UH, respectively;
thus Li has n := 4 + k noncentral 2-chief factors. Next Cq(UL) ::::; Ho, while the
two noncentral chief factors for £ 1 on UL are the two contained in UH, so k ::::; j.On the other hand, Li has two noncentral 2-chief factors on UH, and hence also
two on QH/Ho by 13.7.4.2, and one on 02(Li), so that 5 + j = n = 4 + k. But
now j + 1 = k ::::; j, a contradiction. This contradiction completes the proof of
13.8.33. DLEMMA 13.8.34. (1) Ai::::; UH and Vi::::; UT
(2) m(U;) = l.
(3) 02(Li) iv;, so m(V .. ;)::::; 2.PROOF. By 13.8.33, case (2) of 13.8.8 holds. Then 1 -=/= v; ::::; Ri ~ E4 or
E 8 • By 13.8.4.5, VH/UH is a nontrivial quotient of the 15-dimensional F2H*-
permutation module on H* / L'J..T*, and by 13.8.4.6, v; is quadratic on VH /UH. So
02(Li) i V~', and hence (3) holds. Further Ri. =CH* (U1 Vs), where U1 := CuH (H)and m(UH/Ui Vs)= 2.
Suppose that m(U;) > l. Then as 02(Li) i VC:, m(U~) = 2 and [UH,Ua] =
U1V3, so as [UH,Ua]::::; Ua, UiVs::::; UaVi· Thus UHnU 7 is ofcodimension at most
3inUH,so.
m(D 7 UH/UH) = m(D 7 /(D 7 n UH))= m(D 7 ) - m(U 7 n UH)
= m(UH) - m(U;) __.:. m(U 7 n UH)::::; l.
But [VH,U 7 ]::::; VHnU 7 ::::; D 7 , andhencem([VH/UH,U 7 ])::::; l. However by 13.7.7,
H* is faithful on VH/UH, whereas by G.6.4, u; does not induce transvections with
a common center on any faithful F 2 H* -module. This contradiction shows that
m(U;) = 1, so (2) holds.
Assume that (1) fails. By 13.8.10.1, m(UH/DH) = 1, and we have symmetry
between ')'i and 1' in the sense of Remark F.9.17. Therefore interchanging 1' and
1'1 if necessary, we may assume that Ai i UH, and hence by 13.8.10.2 that U 7
·induces transvections on UH with axis DH· Then as Ai i UH, 13.7.3.7 says
[V-y,DH] ::::; Ai n UH = l. Thus v; induces transvections on fjH with axis DH,
so v; is of rank 1, and hence v; = u;. Then by 13.8.9.2, Vi i U 7 and UH
induces transvections on V 7 /U 7 with center ViU 7 /U-y- Since Vi i U 7 , UH induces
transvections on U 7 /A 1 by 13.8.10.2. However by 13.8.4.5, V 7 /U 7 ° is a nontrivial
quotient of the 15-dimensional F 2 HB-permutation module for G 7 /Q 7 ~ A 6 or (^86)
on HB / £f{TB. Thus as Lf.TB is not the stabilizer of a point in U 7 /Ai, V 7 /U 7 has
a quotient which is the conjugate of U 7 /A 1 by an outer automorphism of G 7 /Q 7.
Therefore as UH induces transvections on U 7 /Ai, it does not induce a transvection
on V 7 /U 7 , a contradiction. This completes the proof of (1) and of the lemma. D
We are now ready to complete the proof of Theorem 13.8.l. As Ai ::::; UH by
13.8.34.1, Ai ::::; Z(Th) for some h EH, and CH (Ai) is a maximal parabolic of H