5.1. PRELIMINARY ANALYSIS OF THE L 2 (2n) CASE(1) Sn KE Syb(K).
(2) Sn LE Syb(L).(3) If K* is of Lie rank 2, then either
(i) S acts on both rank one parabolics of K*, or
(ii) K*S* is Ls(4) extended by a graph automorphism.. 635
PROOF. Note that 02(H) ::; 8 by A.1.6. By 5.1.10, Mk is 2-closed and
02(M'K) E Syl2(K*), so (1) follows. By 5.1.5, B acts on T n L, and hence
T n L::; 02(BT) = 02(MH) = S, so Sn LE Syl2(L), proving (2).
Note by 5.1.10 that B is a Cartan subgroup of K. Thus by inspection of the
groups L 2 (2k) xL 2 (2k), Sz(2k) xSz(2k), (S)L 3 (2k), and Sp 4 (2k) of Lie rank 2 listed
in 5.1.10, either Cr·(B) = 1-so that (i) holds; or K ~ L 3 (4), and Cr.(B*) is of
order 2 and induces a graph automorphism on K*, giving (ii). Hence (3) holds. D
LEMMA 5.1.12. For each X E H(H), Ki lies in a unique Ki(X) E C(X),
K::; K(X) := (Ki(X)T), and one of the following holds:
(1) K = K(X).
(2) Ki < K, Ki/02(Ki) ~ L2(4), and Ki(X)/02(Ki(X)) ~Ji or L2(P), p
prime with p^2 = 1 mod 5 and p = ±3 mod 8.
(3) K/0 2 (K) ~ Sz(2k) and K(X)/0 2 (K(X)) ~^2 F 4 (2k).
(4) K/0 2 (K) ~ L2(2k) and K(X)/0 2 (K(X)) is of Lie type and characteristic
2 and Lie rank 2.
(5) K/0 2 (K) ~ L2(4) and K < K(X) with K(X)/0 2 (K(X)) not of Lie type
and characteristic 2. The possible embeddings are listed in A.3.14.
PROOF. By 1.2.4, Ki lies in a unique Ki(X) E C(X), and the embedding is
described in A.3.12. If Ki < K, then (1) or (2) holds by 1.2.8.2, so we may assume
Ki= K, and hence Ki(X) = K(X) by 1.2.8.1. We may assume (1) does not hold,
so that K < K(X).
As Ki= K, K/02(K) satisfies conclusion (2) or (3) of 5.1.10. In conclusion (3)
of 5.1.10 ask 2: 2, K/02(K) ~ Ls(4) by 1.2.8.4, and then K(X)/02(K(X)) ~ M2s
by A.3.12. However this case is impossible as Tis nontrivial on the Dynkin diagram
of K/0 2 (K), whereas this is not the case for the embedding in M23.
Thus we may assume conclusion (2) of 5.1.10 holds. By 1.2.8.4, K/02(K) is
not unitary, while if K/0 2 (K) is a Suzuki group, then (3) holds by A.3.12. Thus we
may assume K/0 2 (K) ~ L 2 (2k). Then by A.3.12 and A.3.14, (4) or (5) holds. D
LEMMA 5.1.13. Let X E H(H), define k := K(X) as in 5.1.12, and set
D := DL n X. Then either D::; Na(K), or the following hold:
(1) K/02(K) ~ L2(4).
(2) L/02(L) ~ L2(4).
(3) V is the sum of at most two copies of the A5-module.
(4) K::; Ca(Z).
(5) K/02(K) ~ A1, h, or M23.
(6) K(Ca(Z)) = 03 ' (Ca(Z)), and either k = K(Ca(Z)) orK/02(K) ~ A1 with K(Ca(Z))/02(K(Ca(Z))) ~ M2s·