636 5. THE GENERIC CASE: L2(2n) IN .Cf AND n(H) > iPROOF. We may assume D does not act on K, so in particular, D =f. 1. As
k :::J X by 1.2.1, D acts on k but not on K, so K < k and the possibilities for
the embedding of Kink are described in 5.1.12.
If k / 02 (K) is of Lie type of characteristic 2 and Lie rank 2, then K = P^00 ,
where P / 02 ( P) is one of the two maximal parabolics of k / 02 ( K) containing (T n
K)/0 2 (K). Then as D permutes with T, and Tacts on P, also D acts on P, and
hence also on K, contrary to assumption.
Therefore we may assume that case (2) or (5) of 5.1.12 holds. Let De :=
CD(K/02(K)). Then [De,K] ~ [De,K] ~ 02(K) ~ 02(KT), so De acts on
02 (K02(KT)) = K. Thus De< D.
Set (KTD) := KTD/CkrD(K/02(K)); then 1 # D ~ (KTD) ~ Aut(K).
If D acts on K with preimage K+, then D acts on K = K'f, contrary to our
assumption; thus we may also assume that D does not act on K, and in particular
that D 1:. B and so D* =f. 1.
Suppose that case (2) of 5.1.12 holds. The case KJ. ~ L 2 (p) can be handled as
in the case K* ~ L 2 (p) below, so take KJ. ~Ji. As Ki < K, B ~ E 9 is a Sylow
3-subgroup of N f((T n K). Recall B normalizes D, so we may embed B* D* in a
Hall 2'-subgroup E* ~ (Frob 2 i)^2 of Nk• (T* n K*). Now D* is cyclic as D ~ DL.Also D permutes with T, so D is invariant under Nr• (E). But Nr• (E) = (t)
is of order 2, where t interchanges the two components of K, so D* is diagonally
embedded in K. Then as D is cyclic and B-invariant, 01(D) = 1. So D ~ B,
contradicting an earlier reduction. Therefore case (5) of 5.1.12 holds, establishing
(1).
By (1), B ~ B* is of order 3. It remains to consider the corresponding possi-bilities for K* in A.3.14. Furthermore the possibilities of Lie type in characteristic
2 in case (1) of A.3.14 were eliminated earlier.Suppose first that K is not quasisimple. Then by 1.2.1.4, K /O(K*) ~ SL2(P)
for some odd prime p. Let R be the preimage in T of 02 1, 2 (K). As DT =TD, D
centralizes R, and so acts on Ck(R)^00 =:KR; notice KR< k as KR/02(K)) ~
SL2(p). Similarly K ~KR and Tacts on KR; so as KR/02(KR) is quasisimple, D
acts on K by induction on the order of k, contrary to assumption. Thus we may
assume K is quasisimple.
Suppose K ~ L 2 (p) for some odd prime p. Recall in this case that p = ±3
mod 8, so that BT ~ A 4 ; so as B acts on 1 =f. D ~ Aut(K) and DT = T D*,
we conclude D* = B*, contrary to an earlier reduction. As mentioned earlier, thisargument suffices also when Ki < K, where BT ~ A 4 wr Z 2.
Suppose K ~ (S)LH5). Then K = E(Ck.(Z(T)), and as D is cyclic
and permutes with T, we conclude from the structure of Aut(K) that either
D ~ Ck.(Z(T)) ~ Nk.(K), or K ~ £3(5) and DT is the normalizer in
KT of the normal 4-subgroup E of T n K*. In the former case we contradict
our assumption that D does not act on K; in the latter, B ~ NK (DT) = T*,
contradicting B of order 3. Similarly if K ~ £ 2 (25) then as D* permutes with
T B, from the structure of Aut(K), DT = BT ~ KT*, a contradiction.
Next suppose that [D*[ = [D: De[ is not a power of 3. Then as DT =TD, and
K is not of Lie type and characteristic 2, A.3.15 says that K ~Ji, L 2 (qe), L~(q),
for q a suitable odd prime and e ~ 2. Then comparing these groups to our list of