i4.7. FINISHING La(2) WITH (vG1) ABELIAN 105iunique noncentral Li -chief factor on W contained in R/V. So as R ::; CH(U) by
14.7.19.3, and the first paragraph showed that E/V is the unique :rioncentral Li-
chief factor on R/V contained in U /V, we conclude there are exactly two noncentral
Li-chief factors on R/V. Then it follows from 14.7.22 that (2) holds. D
Let Pc:= Cp(U) and fI := H/U.
LEMMA 14.7.24. Assume case (1) of 14. '7.23 holds. Then
(1) S/R ~Core.
(2) Vi< Z(K).PROOF. As we are case in (1) of 14.7.23, case (i) of 14.7.20.2 holds, so that K
centralizes Pc. By 14.7.20.1, p+ := P/Pc is a 6-dimensional irreducible for H*.
Thus Li has exactly six nontrivial 2-chief factors, three each from U and p+. We
next locate these factors relative to the series Q > S > R > V. By 14.7.20.4, one
of the factors is p+ /(P n Q)+, leaving five in P n Q. By 14.7.23, E =Un R::; P,
and the two factors in E are the factors appearing in V and R/V since case (1) of
14. 7 .23 holds; this leaves three factors to be located in Q / R. Further UR/ R ~ U / E
is the natural module for LiT/02(LiT). Therefore applying H.6.5 as in the proof
of 14.7.22, S/R has one of the structures listed in 14.7.22.1. We will show that Li
has exactly two noncentral chief factors in S / R, so that ( 1) will hold by applying
14.7.22.2 to the possibilities in 14.7.22.1.
First U / E is the only factor in S / R contained in UR/ R. This leaves just the
two factors from (P n Q)+ to be located in Q/ R. Now using (1) and (3) of 14.7.19,
[Q,SnP]::; RnP::; Cp(U) =Pc, so that (SnP)+::; Cp+(Q) =:At. Observe
m(At) = 2 by applying the duality in 14.7.19.1 to Co(Q) =Vin view of 14.7.16.3.
By 14.7.16.4, Ua ::; S, and by 14.7.16.2, U~ = Z(T*), so again applying 14.7.19.1,
[P+, Ua] = At ::; (Sn P)+. Hence At = (S n P)+ is of rank 2, so that Li has
exactly two noncentral chief factors on S / R, given by At and UR/ R. As indicated
earlier, this completes the proof of (1).
Define Si as the preimage in S of Soc(S/R). Then Si/R ~Vas S/R ~Core
by (1), so that Si/R = [Si/R,Li]. Observe U i Si since Sis generated by the
L-conjugates of U, so we conclude from the proof of (1) that the noncentral Li-
chief factor in Si/ R comes from At rather than from UR/ R. Thus Bi = PiR,
where Pi := P n Si and P{ = At is of rank 2. So setting Ci := Pc n Si,
CiR/R = Cs 1 ;R(Li) has rank 1. But Co(Li) = 1, so Ci i U, and hence 6i i= 1.
Next as we are in case (i) of 14.7.20.2, P::; K::; CH(Pc) ::; CH(6i), so that
UnR = E, so [C 1 U,P]::; E. Since K centralizes 6i, K normalizes CiU and hence
also [CiU, P], so as K is irreducible on U, we conclude [Ci, P] ::; Vi-that is, P
centralizes C/i. Let Di be the preimage of C 01 o(Li). As P centralizes {\tf, by
Coprime Action we have an Li-module decomposition CiU = Di x U, and then
Li = 02 (Li) centralizes Di. In particular Di ::; Z(P), and hence Di ::; Z(K) by
14.7.21.2. As 6i i= 1, Vi <Di, so (2) is established. D
LEMMA 14.7.25. Vi< Z(K).
PROOF. In case (1) of 14.7.23 we obtained this result in 14.7.24, so we may
assume we are in case (2) of 14.7.23. The proof proceeds much as did the proof of
14.7.24.2, except we analyze P/C rather than P/Pc, where p_ :=[Pc, Li], and