1154 15. THE CASE .Cr(G, T) = f/Jwhere N := m(OvE(B)/OvE(A)). Therefore m(A) ::'.'.'. m(B) + N and hence
2m(A) ::'.'.'. 2m(B) + 2N = m(VE/OvE(A)) + N.
Furtherm(VE/OvE(A)) ::'.'.'. m(VE/OvE(A)) = 2m(VE,i/OvE)A)),
so m(A) ::'.'.'. m(VE,i/OvE 1 (A))+ N/2 with N ::'.'.'. 0, and hence A is an FF*-offender
for the FF-module VE,l·
Next m 3 (GE) ~ 2 since GE is an SQTK-group, and GE ~ 83 x CE, so
m 3 (0E) ~ 1. Therefore by Theorem B.5.1, j ~ L 2 (2n), L3(2m), m odd, or 85,with VE,1/0vE 1 (I) the natural module or the sum of two natural modules for
L3(2m). As GE = IME, VE = (11^1 ), and as V ~ Z(SE), VE = [VE,IJV and
Oj(V) = Oj(V#), where v# -I= 1 is the proj~ction of v on VE,l in the decomposi-
tion of VE. Also v# ~ OvE,l (SE), and by (2), Nj(J(8E)) :::; CE n ME :::; OaE (V),so Nj(J(8E)) :::; Oj(V#)· Using the structure of J(8E) from Theorem B.4.2 we
conclude that j ~ 83 , 85, or L 3 (2). But in the last two cases, as V# :::; Z(SE),
. 031 (01(V)) -I= 1, contradicting 15.3.49.5.
Therefore j 0 ~ 83 and m([VE, 1 ,I]) = 2. At this point, we drop the temporary
assumption that QE = Or(VE)·
By a Frattini Argument, I= Io01CVE), while 01(VE) :::; N1(V) :S ME. Thus as
GE= MEI, GE= MEio, so IGE : MEI = 3. Therefore 02 ,^3 (GE) = 02 ,^3 (0a(V))
is normal in M and GE, so 02 ,^3 (GE) = 02 ,^3 (0M(V)) = 1 as GE 1. M E M.
Then as OM(V) is a 3^1 -group by 15.3.49.4, (1) holds. Hence ME = YETE, so as
IGE: MEI= 3, (3) holds. Further XE= [XE, J(R)] since J(TE) = J(R) by (2), soGE/QE ~ 83 x 83 since YETE/02(YETE) ~ 83 and R:::; 02(ME)· This completes
the proof of 15.3.67. DNext Z is contained in exactly two totally singular 4-subgroups E and F :=
E^8 of V, wheres E 8 - TE. Observe TE =Te = Tp acts on Yp := YJ; with
TE E 8ylz(YFTE), and Y = YEYF02(Y). Let G1 := YpTE, Gz := XETE, and
Go := (G1, Gz). Set Li:= 02 (Gi) and Qi:= 02(Gi) for i = 1, 2. Thus Gi/Qi ~ 83
and TE= G1 n G2 E 8yl2(Gi)·
LEMMA 15.3.68. (1) Go:::; Na(YE)·(2) V:::; Z(Qo), where Qo := 02(Go).
(3) TEE 8ylz(Mo) for each Mo E M(Go).
(4) (Go, G1, G2) is a Goldschmidt triple.
(5) Qo = 031 (Go).
PROOF. By construction, YE ::::1 YTE, so G1 acts on YE. By 15.3.67.3, G 2
acts on YE. Thus Go = (G1, Gz) acts on YE, establishing (1). By 15.3.66.5,
V = [V, YE], so V :::; 02(YE) and hence V :S Qo by (1). Set R := Or(V) as in
15.3.67.1. Then Qo :::; Q1 n Qz = R = OrE(V), so V:::; Z(Qo). Hence (2) holds.
Next let Mo E M(Go). As Na(TE) :::; M by 15.3.67.2, if TE ~ 8yl2(M 0 ) then we
may take T:::; Mo. But then YT= (Yp, T) :::; M 0 , so XE:::; Mo= M = !M(YT)by 15.3.7, contrary to 15.3.67.3. Hence (3) holds and TE E 8ylz(G 0 ), so (4) holds.
Let P := 031(Go)). By F.6.11.1, P is 2-closed with TE n P = Qo, so P