15.1. INITIAL REDUCTIONS WHEN .Cf(G, T) IS EMPTYBy (1),
02 (Cc(V(Mc)))::::; 02 (Cc(Z))::::; 02 (Mz),and then by (*) and (**),
02 (Mz) = 02 (NMnMz(V(Mc)))0^2 (GMc(V(Mc)).
Next from (2), 02 (M n Mz) = 02 (GM(V)), so
02 (Mz) = 02 (GM(V))0^2 (0Mc(V(Mc)),1093and hence 02 (Mz) is T-invariant. Therefore as Sis Sylow in Mz and normal in
T, both 02 (Mz)S = Mz and 02(Mz) = Q1 are also T-invariant, completing the
proof of (3). Then as 02 (Cc(Z)) ::::; Mz and Mc= !M(Gc(Z)), (7) holds. Since
G(G, Q1) ::::; Mc, G(G1, Q1) = Mz = Na 1 (Q1) and Q1 E B2(G1). Then we easily
verify Hypothesis C.2.3 with G1, Q1, Mz in the roles of "H, R, MH", so that (5)
holds. Finally for any 1 f z E Z, Mc E M(Gc(z)), so (6) follows from 1.1.6 appliedto G1, Mc in the roles of "H, M". D
LEMMA 15.1.19. (1) O(G1) = 1.
(2) If K = 02,21(K) is an Mz-invariant subgroup of G 1 with F*(K) = 02 (K),
then K::::; Mz.
(3) 02,F(G1)::::; Mz.
(4) If Mz::::; H::::; G1 with 02,F*(H)::::; Mz, then H = Mz.
(5) Ooo(G1)::::; Mz.
(6) There exists LE C(G 1 ) with L/02(L) quasisimple and L 'f:. Mz.
(7) For L as in (6), 02 (NM(Z1))V2 acts on L and [L, V 2 ] f 1.PROOF. Observe that V2 = [V 2 , 02 (CM(Z 1 ))] by construction in Notation
15.1.16, so V2 centralizes O(G 1 ) by A.1.26.1. Also by construction, 1 f ZnZ 1 Z 2 =
Zn Z1 V2 =: Z+, so that Z+ centralizes O(G1). Now by 15.1.18.6, we may apply
1.1.5.2 with any involution of zt in the role of "z"' so (1) follows.
Assume Kz is a counterexample to (2). Then K is Mz-invariant and S E
Syh(G1) by 15.1.18.4, so 02(K) ::::; 02(KMz) ::::; S ::::; Mz, and hence 02(K) ::::;
02 (Mz) = Q 1 , so that Q 1 E Syh(KQ 1 ). Then by 15.1.18.5 and C.2.5, there is an
A 3 -blockX of KwithX i Mz. Let Y := 02 (Mz); then [Y,X]::::; 02(K)::::; Nc(Y),
so X normalizes 02 (Y0 2 (K)) = Y. However as Mc = !M(MzT) by 15.1.18.7,
X ::::; N c (Y) ::::; Mc, contrary to the choice of X. This contradiction establishes ( 2).
By (1), F*(02,F(G1)) = 02(02,F(G1)), so (2) implies (3).
Assume the hypotheses of (4). Then Q 1 = 02 (H) by A.4.4.1 with Mz in the
role of "K", so H::::; Nc(Q 1 ) ::::; Mc by 15.1.18.5, establishing (4). By (3), we may
apply (4) with 000 (G 1 )Mz in the role of "H", to obtain (5). Similarly if (6) fails,
then by (3), 02 ,F·(G 1 )::::; Mz, so G1 = Mz by (4), contrary to 15.1.18.2. ·
Finally by 1.2.1.3, 02 (CM(Z 1 )) acts on each L satisfying (6), and hence so does
V2 = [Vi, 02 (CM(Z 1 ))]. Further if V2 centralizes L, then so does Zn Z1 V2 f 1,
so that L ::::; Mc = !M(Gc(Z)), contrary to L i Mz. So V2 is nontrivial on L,
establishing (7). D
Recall J(S) = J(T) by 15.1.17, and S E Syh(G 1 ) by 15.1.18.4. Further
15.1.19.6 shows that there is LE C(G 1 ) with L/0 2 (£) quasisimple and Li Mz,
so the collection of subgroups studied in the following result is nonempty: