i050 i4. L 3 (2) IN THE FSU, AND L 2 (2) WHEN .Cf(G, T) IS EMPTYwhere Bo := C 8 M(L/02(L)). As Bi is Sylow in Li and Bi ::; Be ::; BM with
Be cyclic, Be =Bi. Further if Bo -/= 1, then Bo centralizes some a of order 3 in
Bx which is inverted by r E Ri inverting Bx/Bi. But then m3(BoB1(a)) = 3,
contradicting m3(Hx) = 2.
Thus Bo = 1, so B =Bx ~ K* ~ 3i+^2 and Hx = H031(Hx) with Cx =
Li0 3 ,(Hx). Let W := (zeK), so that WE R 2 (CxRi) by B.2.14. As Li centralizes
z and Li :::] Hx, Li centralizes W, so that Cx/CeK(W) is a 3'-group. Since a
3'-group has no FF-modules by Theorem B.4.2, and Ri is Sylow in CxRi, J(Ri)centralizes W by Thompson Factorization B.2.15. Also as H = Gi = Ca(z) by
14.7.15.3, CeK(W) ::; Cx n H ::; Li02(H), so CeK(W) = Li02(Cx). Then
Baum(R1) = Baum(CR 1 (W)) = Baum(02(CxR1)) by B.2.3.5. Therefore Cx ::;
Na(BaumR 1 )::; M by 14.7.16.7. Then by 13.3.8 with Lin the role of "K", 02 (Cx)
is a {2, 3}-group; so as Bi is Sylow in Cx, 02 (Cx) =Li. Thus H = HCx = Hx E
M, and (1) is established.
Let Z 0 := Oi(Z(P)) and assume (2) fails, so that Zp := [Zo,K]-/= 1. Now Zo E
R 2 (K), so Zo = Zp x Cz 0 (K) by Coprime Action. But U 1:. Zo by 14.7.20.1, so Zp
is a faithful irreducible of rank 6 for K* by 14.7.20.2. Hence Zp E R 2 (KRi) is not
an FF-module for K* Ri by Theorem B.5.6; so as Ri E Syl2(KR1), Baum(Ri) :::!
KRi by Solvable Thompson Factorization B.2.16 and B.2.3.5. Thus Baum(R1) :::]
KT= H, contradicting 14.7.16.7. DLEMMA 14.7.22. (1) R/V is isomorphic as an LT/Q-module to one of: the
dual of V; the 6-dimensional core of the permutation module on L/NL(Vi), which
we will denote by Core; the direct sum of the 8-dimensional Steinberg module with
either Core or the dual of V; or the Steinberg module.
(2) Li has three noncentral chief factors on the Steinberg module, two on Core,
and one on the dual of V.
PROOF. As E/V is the natural module for LiT/02(L1T) and R = (EL), (1)
follows from H.6.5. Part (2) follows from H.6.3.3 and H.5.2. D
LEMMA 14.7.23. E =Un R::; P, and either
(1) Case (i) of 14. 7.20.2 holds, R/V is isomorphic to the dual of V as an
L-module, and E /V is the unique noncentral chief factor for Li on R/V; or
(2) Case (ii) of 14. 7.20.2 holds, and R/V ~ Core.
PROOF. By (3) and (4) of 14.7.19, Sis nonabelian while R::; Z(S); so U f:. R
as S = (UL). Then as Li is irreducible on U/E and R = (EL), E = Un R.
Further E = [E,Li] in view of 14.7.16.1, so E::; P. Thus the noncentral Li-chief
factors of R contained in U are the two in E, so E /Vis the unique noncentral chief
factor on R/V contained in U /V. Therefore if case (i) of 14.7.20.2 holds, then as
R::; Z(S) ::; CH(U), E/V is the unique noncentral Li-chief factor on R/V, and
hence R/V is dual to V by 14.7.22, so that (1) holds.
Thus we may assume instead that case (ii) of 14. 7.20.2 holds. Then H has a
unique noncentral chief factor Won CH(U)/U, and H is faithful and irreducible
on W of rank 6. Now [R, Q] = V ::; U by 14.7.19.2, so that [Rn W, Q] = 1,
and hence m( R n W) ::; 2 from the action of Q on the 6-dimensional faithful
irreducible W for K*. As Ua::; S by 14.7.16.4, [Q, Ua]::; R by 14.7.19.1. Therefore
as CH(U) ::::; Cr(V) = Q, [W, Ua] ::; Rn W. Thus as Li acts nontrivially on [W, Ua]
in the 6-dimensional module W, we conclude that Rn W has rank 2, and is the