15.3. THE ELIMINATION OF Mf/CMr(V(Mr)) = S 3 wr Z 2 1153(4) GE/QE ~ S3 x S3 and XE= [XE, J(R)].
PROOF. By 15.3.5.2, if A E AtT) with A i. Cr(V) = R, then either A ::::; Mi
or A = S. Therefore by 15.3.66.3, J(TE) = J(R), so that BE = Baum(R) byB.2.3.5. Hence C(G, BE) ::::; M as M = !M(NM(R)) by A.5.7.2. In particular
Na(TE)::::; M, so as TEE Sylz(ME), TEE Sylz(GE), and hence (2) holds.
Let QC:= 02(Mc) and Pc:= Qc n GE. By 15.3.49.2, QC# 1, so as CJc :::;I T
while Z(T) is of order 2 and lies in TE by 15.3.66.3, Z(T) ::::; Pc and YE= [YE, Pc]·
As Pc is (Mc n GE)-invariant, P:;'E = P"{E since GE= YETECa(E) by 15.3.66.6;
thus as YE= [YE,Pc], YE= 02 ((P:;'E)) :::;I GE. So as V = [V,YE] by 15.3.66.5,VE= [VE, YE]· Further v^0 E = VYETECa(E) = vca(E) s;;; VMc, so that VE::::; VMc·
Thus VE is abelian since VMc is abelian by Theorem 15.3.50.
Let SE := 02(YETE) = CyErE(E) and CE:= Ca(E). Then SE= Cr(E) E
Sylz(CE) by (2). Let GE:= GE/QE. Then GE= YE(i") x CE, where TE Pc - SE
and YE(i") ~ S3. As T tj. SE and E ~ E4, CE(T) = Z. As GE= YE(T)CE andYE(T) acts on V, VE= (VGE) = (VcE).
Let GE := GE/E. Now [V, SE] = E from 15.3.66.1, so QE centralizes VE as
QE::::; SE. Then as we saw GE= YE(i) x CE and YE(i) ~ S 3 ,
VE = VE , 1 E9 VE , 2
is a CE-invariant decomposition, where VE,1 := CvE(T), and VE,2 := CvE(TY) for
1-:/=-y EYE. Thus V}i, 1 = VE,2·
Let I := J(CE)· By (2), J(TE) ::::; Cr(V) ::::; SE, so that J(TE) = J(SE) by
B.2.3.3. Then GE:= INaE(J(TE)) = IME by a Frattini Argument and (2), so as
GE i. M, we conclude Ii. M. Thus j-:/=-1.
Let Io:= N1(Cr(VE)). In order to determine the structure of I 0 , temporarily
replacing GE by NaE(Cr(VE)) if necessary, we may assume that QE = Cr(VE)·
We will drop this assumption later, once we have determined I 0 , and then complete
the proof of (1) and (3).
Let GE:= GE/CaE(VE)· Now (T,QE)::::; TE, so <P((f,QE))::::; <!?(TE)= 1,
and hence <P((T,QE)::::; Ca(V). We saw earlier that i centralizes CE, so CE acts
on (T, QE)· Then as VE = (V^0 E), we conclude that <!?( (T, QE)) ::::; Ca(VE), and
hence <!?( (f, QE)) = 1. Therefore as QE centralizes VE, QE induces a group of
transvections on VE,1 with center CE(T) = Z. Next [YE, QE]::::; 02(YE) = CyE(V),
so as [YE,QE] :::;I GE, [YE,QE]::::; CaE(VE), and hence [YE,QE] = 1. Then as
YJk, 1 = YE,2, CQE (VEJ = CQE (VE)· Hence as QE induces a group of transvections
on VE,l with center Z, we conclude m(VE/CvE(W)) = 2m(W) for each W::::; QE.
As j-:/=-1, there is A E A(TE) with .A-:/=-1. Let B := AnQE and D := CA(VE)·
Then since CvE (A)= An VE as A E A(TE),
m(.A) + m(B) + m(D) = m(A) ~ m(DVE)
~ m(D) + m(VE/(A n VE))= m(D) + m(VE/CvE(A))
so that
m(A-) +m(B) ~ m(VE/CvE(A)).
Further using an earlier remark with Bin the role of "W",
m(VE/CvE(A)) = m(VE/CvE(B)) + N = 2m(B) + N,