13.6. FINISHING THE TREATMENT OF A 5 929PROOF. Let U := [z, X] =/:- 1 and suppose that [U, L~] = 1. Then [X, L~] ::;
Cx· (U) ::; Cx• (z) by Coprime Action, so
1 = [X*,L~,z] = [z,'X*,L~],and hence by the Three-Subgroup Lemma, [L~, z, X] = 1. Then X centralizes
(v)[z,L~] which contains z, contrary to our choice of X*. D
LEMMA 13.6.12. X <:;;; Xz.
PROOF. If X* E X with [z, X*] = 1, then as X* <l G~, X* centralizes
(z^0 ~) = Uv, contrary to X* =/:-1. DLEMMA 13.6.13. No member of X is solvable.
PROOF. Suppose X* E X is solvable, and choose X* minimal subject to thisconstraint. By Theorem B.5.6, X = 02 (H) for some normal subgroup H* of
G~ with H = J(H) = HJ. x · · · x H; where Hi ~ 83 and s ::; 2. Further
UH:= [Uv,H] = U1 E9 · · · E9 Us, where Ui :=[UH, Hi] is of rank 2. By minimality
of X, T; is transitive on the Hi, so XT; is irreducible on UH. Now [z, X] =/:- 1
by 13.6.12, so Ux =UH as X*T; is irreducible on UH. By 13.6.11[UH,L~]=/:-1,
so the projection of L~ on H with respect to the decomposition H x Ca~ (U x) is
nontrivial. Then as Lv is Tv-invariant, it follows that L~ = [L~, J(T)*] = 02 (Hi)
for some i, contrary to 13.6.10. DWe conclude from 13.6.13 that F(J(Gv)) = Z(J(Gv)). Then by 13.6.9 and
Theorem B.5.6, J(Gv)* is a product of components of G~. By C.1.16, J(Tv)*normalizes the components of G~. Thus there exists K+ EC( Gv) such that K.'j.. is a
component of G~ and K.'j.. = [K.'j.., J(Tv)]. Thus (K;Tv) is normal in G~ by 1.2.1.3,
and so lies in X. Hence (K;Tv) E Xz by 13.6.12.
Let Yz consist of those KE C(Gv, Tv) such that K* /0 2 (K*) is quasisimple and
(K;T:;) E Xz. By the previous paragraph, K+ E Yz, so Yz is nonempty. Observe
that if KE Yz and Ko E C(Gv, Tv) with K 0 = K*, then Ko E Yz·For K E Yz, let K_ := (KTv), WK := (zK_Lv), and set (K_LvTv)+ :=
K_LvTv/CK_LvT)WK)· Since F*(Gv) = 02(Gv), WK E R2(K_LvTv) by B.2.14.
In the remainder of the prnof of Theorem 13.6.1, let KE Yz·
Then K+ is a quotient of K* /0 2 (K*), so K+ is also quasisimple, and WK is an
FF-module for K±Tv+· Also the action of K±L:!;Tv+ on WK is described in Theorem
B.5.6.LEMMA 13.6.14. K is Tv-invariant, K* E Xz, and UK= [WK,K].
PROOF. Assume 13.6.14 fails. By 1.2.1.3, K_ = KKt for some t E Tv -
NTv (K), and comparing the list of groups in 1.2.1.3 to that in Theorem B.5.6,
K+ ~ L2(2m) or L3(2). Then by 1.2.2, Lv ::; K_. Since Lv is Tv-invariant with
Lv/02(Lv) of order 3, L-:/; is diagonally embedded in K±, K+ is not L3(2), and
K±T;J is not 85 wr Z 2. Therefore by Theorem B.5.6, UK_ = UKUk, wh~reUK:= [WK,K] and UK/CuK(K) is the natural module for K+ ~ L 2 (2m). Thus
by E.2.3.2, Baum(Tv) is normal in the preimage B of the Borel subgroup B+ of
K± normalizing (Tv n K_)+. But 8 = Baum(Tv) by 13.6.4 and NB(8) ::; Mv