9.4. ELIMINATING n(H) = 1 7359.4. Eliminating n(H) = 1
As we just showed n(H) =f. 2, n(H) = 1 for all H E H*(T, M) by 9.2.5. This
makes weak closure arguments effective, once we obtain restrictions on the weak
closure parameters r and w. •
Define VN and LN as in 9.1.2.4, and let UN:= VJ. By 9.1.2.4, UN= [V,LN]
and UN/VN is the natural module for LN/02(LN) ~ L 2 (4). For v E vtf, setGv := Ca(v).
PROPOSITION 9.4.1. LN <l Gv.
PROOF. Assume the lemma fails. Then Gv i M. We can assume Tv .-
Cr(v) E Syb(CM(v)), and then by 3.2.10.4, Tv E Syl2(Gv)· By 1.2.4, LN :s:; Lv E
C ( Gv) with Lv described in A.3.14, and Lv ::::! Gv by ( +) in 1.2.4 applied to Tv.
We are done if LN = Lv, so assume LN < Lv; thus Lv i M.
We claim that LvTv E He. Suppose first that Lv is quasisimple. As v E
[V, LN] ::; LN ::; Lv, v E Z(Lv), so the multiplier of Lv/Z(Lv) is of even order.
Also Cv(Lv) ::; Cv(LN) = VN, so m2(Aut(Lv)) ;::: m(V/VN) = 6. Inspecting the
lists of A.3.14 and I.1.3 for groups with an automorphism group of 2-rank at least
6, we conclude Lv/Z(Lv) ~ G2(4). But then by I.1.3, Z(Lv) is of order 2, so (v) =
CvN(Lv) and hence m2(Autv(Lv)) = 7 > m2(Aut(G2(4)), a contradiction. Thus
Lv is not quasisimple. As z EUN= [UN,LN] and Cr(02(LvTv))::; Cr(v) = Tv,we conclude using 1.2.11 that LvTv E He.
As LvTv E He, it follows from B.2.14 that U := (zLv) E R2(LvTv)· Notice
using 9.1.2.4 that U contains UN and Vi. Set (LvTv)* := LvTv/CLvrJU).
We next claim that L~ = Ljy, so assume otherwise.
Suppose first that J(T) i Ca(U). Then [L~, J(T)*] =f. 1 and U is an FF-module
for L~T;. If Lv appears in case (c) or (d) of 1.2.1.4 then 000 (Lv)* is a 3'-group,so by B.5.6, [0 00 (L~), J(T)] = 1. Therefore as [L~, J(T)] =/=- 1 and L~/0 00 (L~)
is quasisimple, L~ = [L~, J(T)*]. On the other hand, if Lv/02(Lv) is quasisimple,
then so is L~ = [L~, J(T)]. Thus in any case, L~ = [L~, J(T)] is quasisimple.
Now L~ appears in A.3.14 and B.5.1, and hence as in a previous argument isSL3(4), Sp4(4), G2(4), or A1. Further by B.5.1 and B.4.2, [U,Lvl/C[U,LvJ(Lv)
is either the natural module or the sum of two natural modules for L3(4). As
v E .[UN,LN], v E C[U,LvJ(Lv)· He,nce the 1-cohomology of the natural module
is nontrivial, so that by I.1.6, L~ ~ Sp4(4) or G2(4), and [U,Lv] is a quotient
of a 5-dimensional orthogonal space or the 7-dimensional Cayley algebra over F 4,respectively. Further Ljy = P^00 for some maximal parabolic P of L~. Then
Cu(0 2 (Ljy)) = Cu(02(P*^00 )) contains UN, which does not split over VN, and
v E CvN (L~). This is impossible, since from the structure of these two modules,
Cu(02(P)) = Cu(L~) EB [Ou(02(P)), P*].
Therefore J(T)::; Ca(U). By a Frattini Argument, L~T; = NLvrJJ(T))*, so
as Na(J(T)) ::; M by 3.2.10.1, L~ = Ljy, completing the proof of our second claim.
In particular as LN /0 2 (LN) is simple and U is 2-reduced, the second claim says
L~ = Ljy ~ L2(4); hence 000 (Lv):::; CLJU). Therefore as Lv i M, CLv (U) i M,so case (c) or (d) of 1.2.1.4 holds. In the notation of chapter 1, there is at least
one prime p > 3 with 1 =f. X := Bp(Lv)· Then Xis characteristic in Lv and hence