1142 15. THE CASE .Cr(G, T) = 0
(4) 02(H*) = 1.
PROOF. By 15.3.46.5, Na(Zs) ::::; M = Na(V), so that part (c) of Hypothesis
F.9.1 holds. Let L1 := 02 (Cy(Z)). By 15.3.7, CM(Z) = TCM(V), so L1::::; CM(V).
Now part (b) of Hypothesis F.9.1 holds as Zs ~ T and Zs is of order 2. Further
Zs::::; Z(T). Part (d) holds as M = !M(YT) by 15.3.7.
We next establish part (a) of F.9.1. As Ca(Zs) ::::; M by 15.3.46.5, and
CM(Z) = TCM(V), Ca( Zs)= CM(V)S, so that using Coprime Action,
x := 02 (kercH(Zs)(H))::::; CM(V),
and hence [X, Y]:::; Cy(V). In case (1) of 15.3.7, Cy(V) = 02(Y); thus [Y,X] :S
02 (Y) and L 1 = 1 so L 1 T ::::; H. In case (2) of 15.3.7, Cy(V) = 02,z(Y), and
L 1 :::; Cy (V) :::; H by hypothesis. If X is a 3' -group then again [Y, X] :S 02 (Y)
as Aut(Y/0 2 (Y)) is a {2,3}-group. If Xis not a 3'-group then as 02 (Cy(V)) =
B(CM(V)) by 15.3.7, [Y, X] ::::; Cy(V) ::::; X02(Y). Thus in any case (Y, X] :S
X0 2 (Y), so as X ~ XT, YT normalizes 02 (X0 2 (Y)) = X. It follows that
X = 1, as otherwise H :S Na(X) ::::; M = !M(YT) by 15.3.7, contrary to H E
H(T,M). Thus kercH(Zs)(H) is a 2-group, and hence lies in QH. This completes
the verification of part (a) of F.9.1.
Finally under the hypothesis of part (e) of F.9.1, V^9 :S W 0 (Na(V)).:::; Ca(V)
by 15.3.46.4, so part (e) holds. This completes the verification of (1). By (1), we
may apply F.9.2 to obtain the remaining conclusions of 15.3.48. D
The next result eliminates case (2) of 15.3.7; in particular Lemma 15.3.48 ap-plies thereafter to all members of H(T, M).
LEMMA 15.3.49. (1) 02 (MH):::; CMH(V).
(2) Z =[Zs, 02(Mc)].
(3) Case (1) of 15.3. 7 holds, so Y/02(Y) ~ E 9 , 02 (Y) = Cy(V) = Cy(Z),
and Y = 03 ' (M).
(4) CM(V) and MH are 31 -groups.
(5) Na(Zs) = NM(Zs) is a 3'-group.
PROOF. Part (1) follows since Mn Mc= CM(V)T by 15.3.7.
Since Cy(V) :S Ca(Z) = Mc E H(T, M), enlarging H ir necessary, we may
assume when case (2) of 15.3.7 holds that H contains Cy(V), so that 15.3.48
applies to H.
Let Uc:= CuH(QH)i we claim:
. (a) 02,F• (H) centralizes Uc.
For Uc :S Z(QH ), so as £1(G, T) = 0 by Hypothesis 14.1.5, each member of C(H)
centralizes Uc by A.4.11, and hence 02 ,e(H) centralizes Uc. Also by Coprime
Action, Uc= Cu 0 (02,p(H)) E9 [Uo, 02,F(H)], so as Z :S Cu 0 (Mc) by 15.3.4, and
H :S Mc, it follows that [Uc,02,F(H)] = 1, completing the proof of (a).
Set UH:= UH/Uc. By 15.3.48, H is faithful on UH and 02(H) = 1, while
F(H) centralizes Uc by (a); thus F(H) is faithful on UH, and then also H*
is faithful on UH. In particular UH 1:. Z(QH), so [Zs, QH] = Z by 15.3.48.2, and
hence (2) holds since we may take Mc in the role of "H".
By 15.3.7, (3) holds iff CM(V) is a 3'-group, in which case Mn Mc= CM(V)T