i5.4. COMPLETING THE PROOF OF THE MAIN THEOREM 1165By the claim, J+ is described in Theorem F.6.18. Therefore as Vr is an FF-module for I*, either the lemma holds, or comparing the list of Theorem F.6.18
with that of Theorem B.5.1, we conclude that I* is an extension of L 3 (2), A 6 , A 7 ,
A5, or G2(2), so that C1(Z) is not a 2-group. The latter case contradicts Theorem
15.4.8. D
LEMMA 15.4.22. If F*(If=/= 02(!) then!= KTo, K ~ A5, andI/CT(K) ~ 86 •
PROOF. Assume F*(I) =/= 02(!). By 15.4.20, To E 8yh(Na(02(I))), so we
may apply 15.4.17 to conclude that the hypotheses of 1.1.5 are satisfied and Zn
02(I) = 1. Since Vi = [Vi, Li] ~ E4, Vi centralizes O(J) by A.1.26.1, so 1 =/=
Zn ViZ(I) centralizes O(J) by 15.4.19.3. Thus O(J) = 1 by 1.1.5.2.
If Qi = Q2 then Qi = 02(I); but Z :::::; Qi by B.2.14, contradicting Zn
02(1) = 1. Thus Qi=/= Q2, so (Gt, Tri, Gi) is a Goldschmidt amalgam by F.6.11.2,
and J+ is described in Theorem F.6.18. By F.6.11.1, 03 1(!) is 2-closed, so as
Zn 02 (1) = 1, Zn 03 1(!) = 1 and hence Z ~ z+ is noncyclic; then we conclude
from Theorem F.6.18 that J+ is either L 2 (p^2 ) extended by a field automorphism,
or 87. However F*(I n Mz) = 02(J n Mz) for each z E z# by 1.1.5.1, so that
F*(C1(Cz(W))) = 02(C1(Cz(W))) for W := X, Y by 1.1.3.2. We conclude that
J+ ~ 86. As 031 (I) is 2-closed and F* (I) =!= 02 (I), it follows that I has a component
K with K/0 2 (K) ~ A 6 and then that K = 031 (I) by A.3.18. Thus K = 02 (J) by
F.6.6, so I= KT 0. As E4 ~ Z is faithful on K, Z(K) = 1, so the lemma holds. D
LEMMA 15.4.23. F*(I) =/= 02(1).PROOF. Assume F*(I) = 02(I). Then by 15.4.21, [L2, Li] :::; 02(Li). We
may choose notation so that Li :=Xi, and set Li := X2 and J' := (LiTo,L2).
As [L1, Li] :::; 02(L1) and [L1, L2] :::; 02(L1), we conclude [0^2 (J'), Li] :::; 02(L1)
from F.6.6. However by 15.4.21 and 15.4.22, I' contains an E 9 -subgroup P, with
P n L 1 = 1, since I' 1:. M 1 by 15.4.19.2. Therefore as [0^2 (1'), Li] :::; 02(L1),
m 3 (L1P) = 3, contrary to Na(I) an SQTK-group. D
We are now ready to establish the main result of this section. By 15.4.23, we
may apply 15.4.22, to conclude that Li ~ A4, 02(L1)02(L2) ~ Ds, and 02(L1) n
02 (L2) =/= 1. We may choose Li :=Xi, and set Li := X2 and I' := (Li, L2)· By
symmetry, 02(Li) n 02(L2) =!= 1, so as 02(L1) n 02(Li) = 1,
02(L2) = (02(L1) n 02(L2))(02(Li) n 02(L2)) :::; 02(X) ~ E15.
This is impossible as 02(L1)02(L2) ~ Ds and 02(Li) :::; 02(X).
Since we assume in this subsection that G is not L3(2) or A5, this contradiction
establishes:
THEOREM 15.4.24. Assume Hypothesis 15.4.1. Then G ~ L3(2) or A5.Then combining the main results of this chapter:
THEOREM 15.4.25 (Theorem E). Assume G is a simple QTKE-group, T E
Syl 2 (G), JM(T)J > 1, and Ct(G, T) = 0. Then G is isomorphic to J2, J3,^3 D 4 (2),
the Tits group^2 F4(2)', G2(2)', Mi2, L3(2), or A5.
PROOF. If Hypothesis 15.4.1 holds, the groups in Theorem 15.4.24 appear in
the list of Theorem E. On the other hand if Hypothesis 15.4.1 fails, then there is a
unique member Mc of M(Ca(Z)), so that Hypothesis 14.1.5 holds, and the groups
in Theorem 15.3.1 appear as conclusions in Theorem E. D