608 4. PUSHING UP IN QTKE-GROUPS
PROOF. Notice that the pair I,R satisfies the hypotheses of 4.1.4 for any HE
H(I, M). Since IEμ, there is Hi E M(I) - {M}. By 4.1.4.5, 02(Hi) :::; 02(Hi n
M) SR, while RS R+ S Ca(V). Then VS Oa(02(Hi)) S Hi as Hi EM, so as
F*(Hi nM) = 02(Hi nM) by 4.1.4.6, vs Os1nM(02(Hi nM)) s 02(Hi nM) s
R. Hence VS Z(R), proving (1).
Next NM(R+) acts on R+ and M+, and hence also on [ni(Z(R+)),M+J, so
(2) follows from 4.1.2. Let H E H(I, M). By 4.1.4.3, M+ = H+02(M+), so as
02 (M+) SR+ S CM(V), V = [V,M+] = [V,H+02(M+)] = [V,H+], establishing
(3). " []
4.2. Pushing up in the Fundamental Setup
In this section, we apply the machinery of the previous section in the context
of our Fundamental Setup (3.2.1). Recall from the discussion in Remark 3.2.4 that
under the following assumption, the FSU holds for some V E R-2( (L, T) ):
HYPOTHESIS 4.2.1. G is a simple QTKE-group, T E Syl 2 (G), M E M(T);
and L E £j ( G, T) n M with L / 02 (L) quasisimple.
LEMMA 4.2.2. Hypothesis 4.1.1 holds with M+ :=(LT).
PROOF. By 1.2.1.3, M+ :::l M, and by 1.2.7.3, M = !M(M+T). Further
by 1.4.1.2 02(M+T) = OT(M+/02(M+)) is Sylow in OM(M+/02(M+)), so any
subgroup satisfying the hypotheses on "I" in Hypothesis 4.1.1 contains a Sylow
2-group of M, and hence conjugating in M we may assume T S I. But then
M+T SI, so that M = !M(I), and so Hypothesis 4.1.1 is satisfied. []
HYPOTHESIS 4.2.3. Assume Hypothesis 4.2.1, and setM+ :=(LT) and R+ := OT(M+/02(M+)).
Further assume M S M with M+CM(M+/0 2 (M+)) S M and LT = LM-,
IE μ*(M+, M_), and R := 02(1) SR+.
LEMMA 4.2.4. Assume Hypothesis 4.2.3 and HE H(I, M). Set Ms := HnM,
Ls:= (L n H)^00 , Mo:= (L:Ij{H), and V := [rh(Z(R+)), M+]· Then
( 1) The hypotheses of 4.1.4 and 4.1. 5 are satisfied, with Mo = 02 ( M+ n H) in
the role of "H+ ".
(2) Hypothesis C.2.8 is satisfied.(3) R+ = 02(M+T) = CT(V).
PROOF. By construction V S Z(R+), so that R+ S CT(V). As Lj0 2 (L) is
quasisimple and [L, V] # 1, CM+(V) S 02 ,z(M+), so CT(V) S R+, establishing
(3).
By hypothesis, H E H, so 02(H) # 1 and H is an SQTK-group. Of course
R S H n M = Ms. By 4.2.2 and Hypothesis 4.2.3, the hypotheses of 4.1.4 are
satisfied, so F*(Ms) = 02(Ms) by 4.1.4.6. Thus part (1) of Hypothesis C.2.8 is
established.
By 4.1.4.3, L = Ls0 2 (L), so Ls E C(Ms). Using Hypothesis 4.2.3, LM =
LM-= L^1 ~ LMH ~ LM, so that 02 (M+ n H) = (L:Ij{H) =Mo. Hence Mo plays
the role of "H+" in 4.1.4. Now part (2) of Hypothesis C.2.8 holds by 4.1.4.
Since R2(M+T) S Di(Z(02(M+T))) = ni(Z(R+)), V # l by 1.2.10. Since