10.2. WEAK CLOSURE PARAMETERS AND CONTROL OF CENTRALIZERS 749that mp(Ki) S 1 for each p E 7r(ILI) to the list of embeddings of L 2 (2m) in A.3.12,
we obtain a contradiction.Therefore we may assume instead that F*(Ki) =/= 02 (Ki)· By A.1.26, V =
[V, Lo] centralizes O(Ki), so O(Ki) S Ca(V) S M. Then as 02 (M) S T 1 ,
[02(M), O(Ki)] s 02(M) n O(Ki) = 1, so O(K;) = 1 as M E He. Thus
as Kif0 00 (Ki) is quasisimple, Ki is quasisimple. As Li does not centralize \Ii,
02(Li) i. Z(Ki)· But now each possibile embedding of'Li in Ki in A.3.12 with
02(Li) i. Z(Ki) has mp(Ki) > 1 for some odd prime p dividing ILi, again contra-
dicting our earlier observation. This completes the proof. DAt this point, we eliminate the sixth case of 10.1.1; this will avoid complications
in the proof of 10.2.9.LEMMA 10.2.7. Case (6) of 10.1.1 does not hold. In particular, Cr(V) =
02(LoT).· PROOF. The second statement follows from the first by 10.1.2.3. Assume thefirst statement fails. Then m := m(M, V) = 4 and a := a(M, V) = 2. By Theorem
E.6.3, r := r(G, V);::: m, so r;::: 4 ands:= s(G, V) = 4.
Indeed we show r > 4: For suppose Us V with m(V/U) = 4 and Ca(U) i. M.
If US Cv(x) for some x E iVf#, then x is an involution and U = Cv(x) ;::: Vi for
i = 1 or 2. But then Ca(U) s Ca(Vi) s M by 10.2.6, a contradiction. Therefore
CM(U) = CM(V), and E.6.12 supplies a contradiction.We observe next that 10.1.2.3 and 10.2.3.2 establish Hypothesis E.3.36. A
maximal cyclic subgroup of odd order in M permuting with T is of order 15, so
n'(Auta(V)) = 4 < r. Finally by 10.2.3.1, n(H) S 2 for each H E H*(T, M).
Therefore by E.3.39.2,2 = s - a S w S n(H) S 2
where w := w(G, V) is the weak closure parameter defined in E.3.23. Thus w =
2. Let A S VY be a w-offender in the sense of Definition E.3.27. By E.3.33.4,
A E A2(M, V). Thus 1 =/= Cv 1 (NA(V1)) s Cv(A), so A acts on Vi. As A E
A 2 (M, V), A centralizes O(M) by E.3.40, so m(A/CA(V1)) s m2(Aut(L)) = 2.
Thus m(VY/CA(Vi)) S w + 2 = 4 < r, and hence V1 S Ca(CA(Vi)) S MY.
Similarly Vi s MY, so Vs MY = Na(VY), contrary to E.3.25 since w > 0. DLEMMA 10.2.8. Assume L ~ L3(2) and Cv 1 (L) =/= 1. Set Q := Cr(V). Then
(1) [Z, L] = 1.
(2) ZQ := fh(Z(Q)) = Zr 1 V, where Zr 1 := D1(Z(T1)) = CzQ(Lo).
(3) L = [L, J(T)], and [Z, H] =/= 1 for each HE H*(T, M).
(4) Set Ui := Ci%(T1), let R;, be thepreimage inT of02(CL;(Ui), R := R1R2Q,
and v2 E U2 - Cv 2 (Lo). Then CL 0 r(v2) ~ A4 x L3(2), R = J(T)Q, Cr(v2) =
(T n L)R 2 Q, and D1(Z(Cr(v2))) = Zr 1 (v2).PROOF. As Zr 1 S Cr(V) = Q S T1, Zr 1 = CzQ(T1). As Zi := CV;(Lo) =/= 1
and V1 is a natural module for L, Zi ~ Z2 by B.4.8.1. In particular Cz(L) =/= l, so
(3) follows from 3.1.8.3, since Hi. M = !M(LoT) = !M(Ca(Cz(Lo))).
By 1.4.1.5, ZQ = R2(LoT) with Q = CL 0 r(ZQ) = CL 0 r(V) and V S ZQ.
By (3), EQ is an FF-module for LoT. As V1 E Irr +(ZQ, L) with Cv 1 (L) =/= l,
by part (1) of Theorem B.5.1, V = [ZQ, Lo], and that for any A E A(T) with
L = [L,A] and A minimal subject to this constraint, As Land ZQ = V1CzQ(A).