15.2. FINISHING THE REDUCTION TO Mr/CMr(V(Mr)) c:: o:t(2) 1109(1) Y1Rc/02(Y1Rc) ~ S3, D10, or 8z(2).
(2) Cy 1 (V) = 02(Y1).
(3) M = !M(Y1T).
PROOF. Assume case (3) of 15.2.1 does not hold. Here we take Y 1 := Y. Then
by 15.2.3.1, Y = [Y, Re] = 02 (Mo) and Cy(V) = 02 (Y), so (2) holds, and (3)
follows from 15.2.3.3. Conclusion (1) follows from the structure of Mo described in
15.2.1.So assume that case (3) of 15.2.1 holds. In case (iii) of 15.2.3.1, we again
choose Y1 := Y, so that Y = [Y, Re] is of order 3. In cases (i) and (ii) of 15.2.3.1,
we choose Yo to be the preimage of a T-invariant subgroup Y 0 of Y of order 3
with Yo= [Yo, Re] of order 3, and set Y 1 := 02 (Y 0 ). In each case Y 1 satisfies (1) by
construction. In case (iii) of ;15.2.3.1, Y 1 = Y ~ Y*, so (2) holds; in the remaining
cases we chose Y1 with Y1 ¥ Y*, so again (2) holds. Finally as Y 1 = [Y 1 , Re],
Y1 i Mc, so (3) follows fromL15.2.5. D
LEMMA 15.2.7. Define Y as in 15.2.3 and Y 1 as in 15.2.6. Then(1) M = !M(YT).
(2) If 1-/= X = 02 (X) :<:; CM(V) is T-invariant, then
(i) Nc(X) :<:; M, and
(ii) if [X: 02(X)[ = 3, then X acts on Y1.PROOF. Part (1) follows from 15.2.6.3 as Y 1 :<:; Y. Assume X satisfies the
hypotheses of (2); to prove t2), it suffices by (1) to show that Y acts on X, and
that X acts on Y 1 if [X : 02 (X)[ = 3. Let M* := M/0 2 (M). As T acts on
X = 02 (X) and Y 1 = 02 (Y 1 ), it suffices to show that Y acts on X, and that
X* acts on Yt if X, Y[ = 3. But as Y :Si M by 15.2.3.2, [X, Y] :<:; Cy(V),
so if Cy(V) = 02(Y), then [X, Y] = 1, and the lemma holds. Thus by 15.2.3.1,
we may assume that case (ii) of 15.2.3.1 holds. Then [X, Y] :<:; Z(Y*) with
Z(Y) of order 3. Thus if X is a 3'-group, then X = 02 (X) centralizes Y* by
Coprime Action, and as before the lemma holds. Finally if X* is not a 3' -group,
then Z(Y) :<:; X as case (ii) of 15.2.3.l holds, so [X, Y] :<:; Z(Y) :<:; X, and
once again Y acts on X. Also if [X: 02 (X)[ = 3, then X = Z(Y) acts on Yi*,
completing the proof. D
LEMMA 15.2.8. If HE 1-i*(T, M), then H/02(H) ~ 83 wr Z2.
PROOF. First H/CH(U 1 ) is described in 15.1.12.3, where U1 is a noncentral
chief factor for Hon UH:= (VH). In particular 02(H/CH(U1)) = 1, so 02(H) :<:;
CH(U 1 ). Recall by 15.1.9.7 that H is a minimal parabolic described by B.6.8;
thus H n M = NH(T n H) by 3.1.3.1, and CH(U1) :<:; H n M by B.6.8.6a. If
CH(U1) > 02(H), then X := 02 (CH(U1)) -/= 1, so that X :<:; CM(V) by 15.1.9.4;
hence by 15.2.7.2, H :<:; Nc(X) :<:; M, contrary to H E H*(T, M). Therefore
02(H) = CH(U1).
Thus H/CH(U 1 ) = H/0 2 (H) is described in 15.1.12.3. By 15.2.4, H/0 2 (H) is
not 85 wr Z 2 , and the lemma holds if H/02(H) is 83 wr Z2, so we may assume
that H/0 2 (H) ~ 83 or 85, and it remains to derive a contradiction.
Define Re and Y as in 15.2.3, and Y 1 as in 15.2.6. We will verify that Hypothesis
F.1.1 is satisfied with Y1Rc, H, T in the roles of "£1, £2, 8". Most parts are
straightforward, but we give a few details: First Y1Rc/02(Y1Rc) ~ 83, D10, or
8z(2) by 15.2.6.1, while we saw H/0 2 (H) ~ 83 or 85, so that part (c) holds. Next