3.3. NORMALIZERS OF UNIQUENESS GROUPS CONTAIN Na(T) 595.( c) K 0 < K* ~ L3(p) for some odd prime p.
In case (b), D* = 1, contrary to ~n earlier remark. Suppose case (c) holds.
Then from the structure of NAut(K•)(T*), D* :::; Den where D 0 is a cyclic sub-
group of K* of order dividing p - E centralizing T*. Further we saw that K 0 ~
SL2(p) or SL2(P)/Ep2· But now [K 0 ,D*] :::; [K 0 ,D 0 ] :::; O(K 0 ), contradicting
G 0 = (X*T*,X*^9 T*).
Therefore case (a) holds, with K 0 = K* = Y* = F*(B*), and D* acts on
Y*T* = G 0 , so G 0 :SI B* = Y*T* D* :::; Aut(K*). Recall Gt satisfies one of cases
(3)-(13) of F.6.18, but does not satisfy (b). As F*(G*) = K* is simple and Kt is
quasisimple, K* ~Kt /Z(Kt). Examining F.6.18 for groups with T* <Na• (T*),we conclude case ( 4) or (10) of F.6.18 holds. However G2 = Gf, so G2 ~ Gi, ruling
out case (10) of F.6.18, since Gt Z(Kt)/Z(Kt) ~ Gi ~ G2 ~Gt Z(Kt)/Z(Kt).
This leaves case (4) of F.6.18, so we conclude that Gt = Kft ~ L 2 (p), p = ±11
mod 24, and x+T+ ~ D12· As Gt is simple, Gt ~ G 0 = K. Further Aut(K) is
a 2-group, so B = G 0 D = K ~Gt.
Next there is t E T n K with X = [X, t], so X = [X, t] :::; K, and hence
Y = (XD) :::; K as K :Si B. By (1), Y :Si B = YTD, so since K E C(B)
with K = B ~ Gt ~ L2(P), we conclude from 1.2.1.4 that either (2) holds,
or Y/0 2 (Y) ~ SL 2 (p)/Ep2. However in the latter case, by a Frattini Argument,
Y = Op(Y)Yo, where Yo := Ny(T1) and Ti := T n 000 (Y). But then XT and D
act on Ti, so Ti :::; 02(B), whereas Ti 1:. 02(Y). Thus (2) is established.
We saw that XT ~ D 12 , and from (2), NB(T) ~ A4, so (3) follows. Further
we observed earlier that B ~ Gt, so B = (XT, X^9 T) for g E D with g -=J 1.
Let W be a noncentral 2-chief factor of Y, n := m(W) and a:= m([W, X])/2.
Then a is the number of noncentral chief factors for X* on W, so a :::; 'Y· As
B = (XT,XBT), Cw(X) n Cw(X^9 ) = O, son:::; 2m([W,X]) = 4a. On the
other hand, a Borel subgroup of B* is a Ftobenius group of order p(p - 1)/2, so
n ~ (p -1)/2 and hence p:::; 2n + 1 :::; 8a + 1. Thus either a> 4 or p:::; 33, and in
the latter case as p = ±11 mod 24, p = 11 or 13. As neither 11 nor 13 divides the
order of GL 9 (2), we conclude that n ~ 10 and hence a~ n/4 > 2. Thus as 'Y ~a,
(4) holds.
It remains to prove (5), so assume that 'Y :::; 4. Then a :::; 4, so by the previous
paragraph, Wis the unique noncentral 2-chief factor for Y, and m(W) ~ 10. Then
as IT* I = 4, ITI ~ 212 , with equality only if p = 11 and W = 02 (YT), so that
Z(YT) = 1. Therefore parts (a) and (c) of (5) hold. Finally W = U/Uo where
U := [0 2 (Y), Y] and Uo := Cu(Y), and as 02(X) :::; 02(Y) by (3), 02(X) =
[0 2 (X),X] :::; U. Then as U/U 0 is elementary abelian and X:::; Y, <[>(02(X)) :::;
U 0 :::; Z(Y), establishing part (b) of (5). This completes the proof. D
In the next lemma, we eliminate the first occurrence of the shadow of nt (2)
extended by triality.
PROPOSITION 3.3.21. L is not L4(2), eliminating case (1) of 3.3.8.
PROOF. Assume otherwise. Arguing as in the proof of 3.3.19 via appeals to
Theorems B.5.1, B.4.2, and 3.3.18, we conclude:
(a) Either
(1) [V, L] = U 1 EB U 2 , where U 1 is a natural submodule of V and U2 is the dual
of U1, or