14.7. FINISHING La(2) WITH (VG1) ABELIAN 1055contradiction. Hence Po n M = 1, so in particular B f:_ Po. Then as Po contains
each subgroup of order 3 in Cp(Po), P1 := Cp 0 (B) < P 0. Now TL 1 acts on P{, so
as Pi < Po, P1 ::::; M by minimality of H, contradicting Po n M = 1. This completes
the proof of 14.7.27. D
LEMMA 14.7.28. For each HE 1-lz, either O(H*) = 1 or O(H*) =Li.
PROOF. Suppose H is a counterexample. Then by 14.7.27, Op(H*) =/= 1 forsome prime p > 3. But this contradicts 14. 7.6. D
As a corollary to 14.7.28 we haveTHEOREM 14.7.29. Each solvable subgroup of G 1 containing L 1 T is contained
inM.
PROOF. Assume L 1 T ::::; H f:_ M is solvable. Then 1 =/= O(H) = F(H*) =
Li~ Z3 by 14.7.28, and hence IH*: CH·(F*(H*))I::::; 2. Then H = QHL1T::::; M,
contrary to assumption. D
14.7.3. Reducing to 02 (H*) isomorphic to G 2 (2)' or A 5. Let HE 1-lz·By Theorem 14.7.29 and 1.2.1.1, H contains C-components. In this subsection, we
establish restrictions on the C-components of H: For example, 14.7.48 will show
that H contains a unique C-component K, and that H = KT. Then Theorem
14.7.52 will reduce our analysis to the cases where K/0 2 (K) ~ A5 or G2(2)'.
Let KE C(H). By 14.7.28, IO(K*)I::::; 3, so K/02(K) is quasisimple by 1.2.1.4.
Also Ki M and (KT)L 1 T E 1-lz by 14.5.19, and hence:
LEMMA 14.7.30. For each K E C(H), K/0 2 (K) is quasisimple, K j;_ M,
(KT)L{F E 1-lz, and K/0 2 (K) is described in F.9.18.LEMMA 14.7.31. Suppose Ca(Vi)::::; M. Then
(1) Wo(R1, V) :::] LT, so Na(Wo(R1, V))::::; M.
(2) Let U := (VG^1 ) and assume there is Y E 1-le, Ty E Syl 2 (Y), and Vy E
R 2 (Y) with Y/0 2 (Y) ~ 83 , 02(Y) = Cy(Vy), and UB::::; Cy(Vy) for each V.f1::::;
Vy. Then Wo(Ty, V) :::] Y.
PROOF. Observe first that as Ca(Vi) ::::; M by hypothesis, Ca(V2) ::::; Mv by
14.3.3. Thus as Lis transitive on hyperplanes of V:
(*) Ca(A)::::; Na(VB) for each g E G and each hyperplane A of VB.
· Suppose VB ::::; R 1 with VB =/= 1. Then
V = (Cv(A): m(VB /A)= 1),while for each hyperplane A of VB, [Cv(A), VB] ::::; V n VB = 1 by(*) and 14.5.2.
Thus [V, VB] = 1, contrary to assumption. We conclude Wo(R1, V) ::::; Cr(V) =
02 (LT), so by E.3.15 and E.3.16, Wo(R1, V) = W 0 (02(LT), V) :::] LT and also
Nc(Wo(R1, V)) ::::; M = !M(LT). Thus (1) holds.
Assume the hypotheses of-(2), and suppose VB ::::; Ty with [Vy, VB] =/= 1. Then
A := VB n 02 (Y) is a hyperplane of VB, so by (*), [Vy, VB] ::::; Vy n VB, and hence
by transitivity of L on V#, we may take V.f1 ::::; Vy. Then VB ::::; UB ::::; Cy (Vy) by
hypothesis, contrary to assumption. Thus Wo(Ty, V) = Wo(02(Y), V) :::] Y using
E.3.15 just as in the proof of (1). D
.LEMMA 14.7.32. T normalizes each KE C(H), so KL1T E 1-lz·