7.7. MINI-APPENDIX: r > 2 FOR L 3 (2).2 ON 3 EB 3 7057. 7.3. Proof of Proposition 7. 7.2. In the remaining two subsections of the
section, we assume that either
(HO) V+ = Vo and Go i:. M, or
(Hl) V+ = Z1, r > 2, and Gi ¢.He.In each case, we work toward a contradiction. In this subsection, we assume
(Hl) and obtain a contradiction establishing Proposition 7.7.2, and hence also com-pleting the proof of lemma 7.6.1, which depended upon that Proposition. At the
same time, we will prove a lemma 7. 7.6, necessary for the proof of Proposition 7. 7.3.Then in the final subsection we assume (HO) and complete the proof of Proposition
7.7.3, on which various earlier results depended.Under (HO), choose H E H*(L^0 T 0 , M) with H :::; G 0. Under (Hl), choose
HE H(L1T1, M) with H:::; G 1 , and H minimal subject to H ¢.He.
In either case set MH := H n M. As H E H, H is an SQTK-group. SetA := V+ V_; and observe that A = V under (HO), while A is a hyperplane of
V under (Hl). Therefore Ca(A) :::; M under either hypothesis, since r > 1 in
Hypothesis (Hl).
Under Hypothesis (HO) we will prove:
LEMMA 7.7.6. Assume Hypothesis (HO). Then
(1) To E Syl2(H).(2) H = J(H)L^0 To.
(3) F*(H) = 02(H).
We prove lemma 7.7.6 and Proposition 7.7.2 together.First assume just Hypothesis (HO). Since To is Sylow in G 0 , part (1) of 7.7.6
holds. As 02 (L^0 T 0 ) = Q, with To Sylow in both L^0 T 0 and H, we conclude from
A.1.6 that 02 (H) :::; Q. By a Frattini Argument, H = J(H)NH(R), where R :=
Ton J(H) E Syb(J(H)), and J(T) = J(R). Then NH(R):::; M by 3.2.10.8, so as
H i:. M, also J(H) i:. M-and hence part (2) of 7.7.6 follows from minimality of
H.It now remains to prove part (3) of 7.7.6, as well as Proposition 7.7.2. Thus
we assume either (HO) or (Hl), and it remains to show that F*(H) = 02 (H). As
a first step, A.1.6 says 02(M):::; Q:::; T+:::; H, so by 1.1.4.5, F*(MH) = 02(MH)·Next applying A.1.26.1 to L^0 on V = [V, L^0 ], V_ centralizes O(H). Therefore
O(H):::; CH(V_) = CH(V+V-) = CH(A).
Thus given our earlier observation that Ca(A):::; M, O(H):::; O(MH), so O(H) = 1
since MH E He.
It remains to show that E(H) = 1. Thus we may assume that there is a
component K of H. If K:::; M, then K:::; E(MH), contradicting MH E He; thus
K f:. M. By 1.2.1.3, L+ = 02 (L+) :::; NH(K), so L+T+ acts on Ko := (KT+).
Therefore by minimality of H, H = K 0 L+T+.
Next as L^0 acts on K, so does V = [V,L^0 ]. We claim V_ acts faithfully
on K, so assume otherwise; the proof will require several paragraphs. First V+ <
W := CA(K), so as L^0 acts on W, W contains at least one of the five nontrivial
orbits 0 of (e) on V!!. Now 0 = W~ for some 2-subspace W_ of W. Observe
W contains no involution w 2-central in G: For if w is such an involution, then
K:::; E(H) n Gw:::; E(H n Gw), while E(H n Gw) = 1 by 7.7.5.