677By 6.1.24.2, R =f. 1 ass;::: 1. By 6.1.24.4, R is faithful on F(J). Thus by
6.1.24.3, R* is faithful on E(I*), so there is K E C(J) with K/0 2 (K) quasisimple
and [K*, R*] =f. 1. As IKHI S 2 by 1.2.1.3, DL acts on K; further DL n I = 1.
So as R* = [ R*, D L] by 6.1.24.2, R also acts on each member of K H, and hence
[K*, R*] = K*. Let MK := Mn K, and SK := Sn K; then SK E Syl 2 (K) asSE Sylz(J).
We claim that K i. M, so that M'K < K* as CH(U) s Na(V) s M: For
otherwise K S CM(Zs) s Mv ::::; Na(R) using 6.1.7.1 and 6.1.21.5, contradicting
[K*,R*] = K*.
LEMMA 6.1.25. (1) n = 2.
(2) K* ~ Lz(p), p = ±3 mod 8, p;::: 11.
(3) s = 1, so that R/ E is the natural module for Li/ R.
PROOF. First DL normalizes S E Sylz(I), and hence also normalizes S'K E
Sylz(K*). If DK := CvL (K*) =f. 1, then as we saw R* acts on K*, R* = [R*, DK] ::::;C1 (K) by 6.1.24.2, contrary to the choice of K. Thus DL is faithful on K*.
Therefore either
(A) DL is a 3-group, and hence of order 3 with n = 2, or(B) K /Z(K) is described in A.3.15 with Z(K*) of odd order by 6.1.24.4.
Assume for the moment that (B) holds. As DL acts on S'K, it follows from
A.3.15 that one of the following holds:
(a) K* is of Lie type and characteristic 2.
(b) K* is Ji and n = 3 as DL has order 7.
(c) K* is (S)L3(p) and D£ n K* = 1.However in case (c), using the description in A.3.15.3, DL centralizes SK:. As
R = [R, DL] and Out(K) ~ S 3 , R induces inner automorphisms on K*, impos-
sible as 1 =f. R = [R,DL] and DL centralizes SK:. This eliminates case (c).
Now assume for the moment that (A) holds. We check the list of Theorem C
(A.2.3) for groups K /Z(K) in which the normalizer of SK: in Aut(K /Z(K))
contains a subgroup of order 3, and conclude that either K* is of Lie type and
characteristic 2, or K is L 2 (p) withp = ±3 mod 8 or Jz. The case where K ~ Jz
is ruled out by A.3.18 as DL n I= 1.
Next suppose (A) or (B) holds and K* is of Lie type over F 2 k. Then as DL
acts on SK:, either k > 1, or K* is^3 D 4 (2) and DL is of order 7-so that n = 3.
In any case, DL acts on a Borel subgroup B of K containing SK:. Further either
K is of Lie rank 1, in which case we set Ki := K, or K is of Lie rank 2. In the
latter case, as Ki. M, either
(i) DLT acts on a maximal parabolic P* of K with preimage P satisfying
Ki := 02 ' (P) i. M, or
(ii) K* is Sp 4 (2k) or (S)L 3 (2k) and T is nontrivial on the Dynkin diagram of
K*, and we set Ki := K.
In any case, Ki i. M.
Suppose first that B::::; M. Then H 2 :=(Ki, T) E 1-l*(T, M) with n(H 2 ) > 1-