3.3. NORMALIZERS OF UNIQUENESS GROUPS CONTAIN Na(T) 597We have Z :S: [V, L] = V by (e), and T i_ L0 2 (LT) by (f), so V = WEB Wt for
t ET-L02(LT) with W := U 1 the natural module for Land wt dual to W. In
particular Z ~ Z 2 is D-invariant, and we saw D::::; Na(H), so D nor:m:alizes
u := (zH) = (Un W) EB (Un W)t,
with Un W ~ E4. Now H acts as ot(2) on U, so AutH(U) is self normalizing in
GL(U) and AutT(U) is self normalizing in AutH(U); thus we conclude [U, D] = l.Hence [ H, DJ :S: CH (U) = 02 ( H); in particular D centralizes T / 02 ( H), so D acts
on S := TnL02(LT), and hence on Zw := Cw(S), since Zw ::::; U and D centralizes
u.
Let Lw :·= CL(Zw)^00 • Then Lw/02(Lw) ~ L3(2), and Lw has noncentral
chief factors on each of W/Zw, wt, and 02 (Lw). We will now apply earlierarguments to see that (LwS, S) cannot be an MS-pair; then since (MSl) and
(MS2) hold, we can conclude (MS3) does not hold. So suppose (MS3) does hold:
then we may apply C.l.32, and as before one of cases (1)-(4) of C.l.34 holds. Since
we saw there are at least three noncentral 2-chief factors, cases (1) and (2) of C.l.34
are eliminated. As Zw ::::; W = [W, Lw] is a nonsplit extension of a natural quotient
over a trivial submodule, case (3) of C.l.34 does not hold. We've seen m(Z) = 1,
so as IT: SI = 2, m(Z(S)) ::::; 2, and hence case (4) of C.l.34 does not hold. This
contradiction shows that (MS3) fails, so there is 1-=/= C char S with C :::) LwS. But
then C :::) (Lw, T) =LT, while D normalizes S and hence also C, contradicting
3.3.6.b. D
LEMMA 3.3.22. L is not A1, eliminating case (3) of 3.3.8.
PROOF. If L ~ A1 then by 3.3.8 and 3.3.17, [V, L] is the natural module for L.
We adopt the notational conventions of section B.3; that is we regard LT~ S1 as
the group of permutations on n := {1, ... , 7}, [V, L] as the set of even subsets of
n, and take f' to have orbits {1, 2, 3, 4}, {5, 6}, {7} on n. Set e := n - {7}; then
Zv :=Zn [V, L] = (es,6, eo).
Let Lo := CL(eo)^00 • Observe Lo ~ A6 and R := 02(LT) = 02(LoT), with
C( G, R) ::::; M by 1.4.l.l.
Consider any z E Cz(L)eo, and set Gz := Ca(z) and Mz := CM(z). Then
z E Z, so that Gz E ?-le by 1.1.4.6. As Lo :::) Mz, RE B2(Gz) and RE Syl2( (RMz))
by A.4.2.7, so as C(G, R) ::::; M, it follows that Hypothesis C.2.3 is satisfied with
Gz, Mz in the roles of "H, MH". Further by 1.2.4, Lo :S: K E C(Gz)· Now
F(K) = 02 (K) by 1.1.3.1, and m3(Lo) = 2, so K/02(K) is quasisimple by 1.2.l.4
and Tacts on K by 1.2.1.3. Assume Lo < K, so that K i Na(L) = M. Then
C.2. 7 supplies a contradiction, as in none of the cases listed there does there exist
a T-invariant Lo E C(M n K) with Lo/02(Lo) ~ A5. Hence Lo= K:::) Gz· Thus
by A.3.18
Lo= 0
31
(Ca(z)) for each z E Cz(L)eo. ()
Similarly for z E Cz(L), Ca(z) :S: Mas M = !M(LT), so by A.3.18:
L = 031 (Ca(z)) for each z E Cz(L). (**)
Now as Grv,L] (L) = 0, 3.3.7.4 says that V = [V, L]EBCz(L) and Z = ZvEBCz(L).
We claim that Cz(L) = 1, so that V = [V, L] and Z = Zv. Assume otherwise.
Then m(Z) > 2, and Zo := (Cz(L), eo) is a hyperplane of Z, so for each d E D,
1 -=/= Zo n zg. Hence we may choose z E z'/f with zd E Zo. First suppose z E