i5.4. COMPLETING THE PROOF OF THE MAIN THEOREM 1155By 15.3.67.2, J(TE) ~ R, so that J(TE) :Si YFTE, and so Yp acts on X. As Xis
a 3'-group with F*(X) = 02(X), X = Cx(Zo)Nx(J(TE)) by Solvable Thompson
Factorization B.2.16. As Yp acts on X, F = (ZYF) ~ Zo by B.2.14, so P =
Cp(F)Np(J(TE)). But by 15.3.67, Cc(F) and M = YT are {2, 3}-groups, so
P = Qo as Nc(J(TE)) ~ M by 15.3.67.2. Thus (5) holds, completing the proof of
15.3.68. DWe are now in a position to complete the proof of Theorem 15.3.1.
Let Qo := 02(Go) and Go := Go/Qo. By 15.3.68.3 and F.6.5.1, (Gi, TE, G 2 )
is a Goldschmidt amalgam. Since Gin G2 = TE, and Q 3 ,(G 0 ) = Q 0 ~ TE by
15.3.68.5 case (i) of F.6.11.2 holds, so G 0 is described in Theorem F.6.18.Let Vo:= (VG^0 ). By 15.3.68.2, Vo~ fh(Z(Qo)). Also Cc 0 (Vo) ~ Cc 0 (V) = R
is a 2-group by 15.3.67.1, so Qo = Cc 0 (Vo). By 15.3.67.4, XE = [XE, J(TE)] ~
J(Go) =: X, so Vo is an FF-module for Go. Thus examining the list of Theorem
F.6.18 for groups appearing in Theorem B.5.6, and recalling that J(TE) :SJ Gi by
15.3.67.2, we conclude that X 9:! 83, £3(2), A6, 86, A1, 87, A.6, or G 2 (2).
Assume first that X 9:! 83. Then XE = 02 (X), so Z ~ Cc 0 (X), and hence
F = (ZYF) ~ Ca 0 (X). But then XE acts on EF = z1-, so XE~ M by 15.3.46.2,
contrary to 15.3.67.3.In the remaining cases, 02 (X) = 02 (G 0 ) by Theorem F.6.18, so Yp ~ X.
However Qo02(Yp) ~ CrE (V) = R, so 02(YF) centralizes V, while ["V, Qi] = F,
so Qi > 02(YF ). This eliminates the cases Go 9:! £ 3 (2), A6,· A1, or A 6 , so thatGo is 86, 86, 81, or G2(2). As V = [V,Yp] ~ [Vo,X], Vo= [Vo,X]. Thus 02(Yp)
centralizes the 4-dimensional subspace V of the FF-module V 0 = [Vo, X] for X,
so we conclude using Theorem B.5.1 that Go is 86 and m(V 0 ) = 6. But now
N.x(Vi) has a quotient A 5 , whereas Nc(Vi) ~ M by 15.3.45.2, and Mis solvable
by 15.3.67.1.
This contradiction completes the proof of Theorem 15.3.1.15.4. Completing the proof of the Main Theorem
In this section, we complete the treatment of the case £ f ( G, T) empty, and
hence also the proof of the Main Theorem. Our efforts so far have in effect reducedus to the case £(G,T) empty (cf. 15.4.2.1 below).
More precisely, since we have been assuming that IM(T) I > 1, and since The-
orem 15.3.1 completed the treatment of groups satisfying Hypothesis 14.1.5, wemay assume that condition (2) of Hypothesis 14.1.5 fails. Thus in this section, we
assume instead:HYPOTHESIS 15.4.1. G is a simple QTKE-group, TE 8yb(G), and
(1) Lt(G, T) = 0.
(2) Let Z := f!i(Z(T)). Theri IM(Cc(Z))I > 1.The section culminates in Theorem 15.4.24, where we see that £ 3 (2) and A 6
are the only groups which satisfy Hypothesis 15.4.1.We now define a collection of subgroups similar to the set B(G, T) of chapter1: Let ~(G, T) consist of those T-invariant subgroups X = 02 (X) of G such that
XT E 1i(T) and IX : 02(X)I is an odd prime. Let C(G, T) consist of those
XE ~(G,T) such that :3!M(XT).