S.l. PRELIMINARY ANALYSIS OF THE L 2 (2n) CASE 637embeddings of As in A.3.14, we conclude K* ~ Ji. As D f:_ Nc(K) is cyclic, we
conclude that D = [D, B*J is of order 7; hence as D ::::; D L = NL (T n L), n = 3m
for some m. In particular as B does not centralize D, B induces a group of field
automorphisms of order 3 on L/02(L). Further DnK =: D 7 is the subgroup of DL
of order 7. If all noncentral 2-chief factors of Lon V are natural, then Cn(Z) = 1.
If not, then by 5.1.3, mis even so that m = 2s for some s, and the unique noncentral
chief factor is orthogonal; so as 7 divides 238 - 1 = 2n/^2 - 1, [Z, D 7 ] =/=-1. Hence in
any case [Z, D 7 ] =!=-l, so as D 7 ::::; K, [Z, K] =!=-1. Thus (zk) E R'2(K) by B.2.14,
so that K E £-J(G, T). Then K E £j(G, T) by 1.2.8.4. Now by 3.2.3, a suitable
module for K satisfies the FSU. As Ji does not appear among the possibilities for
"L" given in 3.2.6-3.2.9, this is a contradiction.
Thus D is a 3-group, and we have seen D i B, so B D* is a 3-group
of order at least 9 permuting with T. Inspecting the possibilities for K in the
remaining cases of A.3.14, we conclude that K/02(K) ~ A7, A7, h, or M23, and
D is of order 3 and inverted by some t E K n T. (There are groups of order 9 in
J 4 containing B and permuting with T, but each such group acts on K*). Since
D i B and B acts on the cyclic group D, K/02(K) is not A7, establishing (5).
Next K = 031 (X) by A.3.18, so B0 3 (D) ::::; k. Hence as D* is a 3-group,
D = 03(D) X De, with 03(D) =: D3 of order 3 and De= 03 (D).
Now K::::; KE L*(G,T) and D3::::; K::::; K with D3 i Nk(K). Therefore K
satisfies the hypotheses of K, and hence replacing K by K if necessary, we may
assume k =KE L*(G,T).
We next prove (4) by contradiction, so we assume that Ki Cc(Z) and choose
V so that Z ::::; V; this argument will require several paragraphs. By 5.1.7.1,
Baum(T) is not normal in KT, so K = [K, J(T)] using B.6.8.6.d. Set U :=
[(zk), K], so that U E R 2 (KT) by B.2.14 and U is an FF-module for KT by B.2.7.
Then as M 23 and J 2 do not have FF-modules by B.4.2, K/02(K) ~ A7. Hence as
B D is of order 9, B DT* is the stabilizer of a partition of type 3, 4 in the 7-set
permuted by KT, and KT is the stabilizer of a partition of type 2, 5. By B.5.1
and B.4.2, U is irreducible of dimension 4 or 6, with (zk) = UZ = U x Cz(K).
Then from the action of Kon U, [Zn U, K] =/=-1, so by 3.1.8.3, L = [L, J(T)].
Therefore by 5.1.2, V/Cv(L) is the natural module or the As-module for L.
Suppose first that V/Cv(L) is the As-module. Then DL = D = D3 ::::; Cc(Z).
But if m(U) = 6, then B = CBn(Z), contradicting B i D. Hence m(U) =
4. However from the description of FF* -offenders in B.4.2. 7, N k• ( J(T)) is the
stabilizer in K* of a partition of type 3, 4, so J(T) :::;I EDT; while as [V, L] is the
Es-module, J(T) is not normal in DT.
Therefore V/Cv(L) is the natural module. Then J(T) ::::; (T n L)02(LT) by
B.4.2.1, so that J(T) :::;I DT. If m(U) = 6, then J(T) is not normal in D3T using
the discussion of FF*-offenders in B.3.2.4; hence m(U) = 4.
As V/Cv(L) is the natural module, [Z,D 3 ] =!=- 1 and Cz(D3) = Cz(L). Then
as m(U) = 4 and [Z, D3] =!=-l, with UZ = U x Cz(K), Cz(D3) = Cu(K) = Cz(K).
Therefore Cz(L) = Cz(K), so Cz(L) = Cz(K) = 1 as H =KT i M = !M(LT).
Then Z::::; U; so Cz(K) ::::; Cu(K) = 1. Next by C.1.28, either there is a nontrivial
characteristic subgroup C of Baum(T) normal in both LT and KT, or one of Lor
K is a block. As M = !M(LT) but Ki M, Lor K is a block.