650 5. THE GENERIC CASE: L2(2n) IN .Ct AND n(H) > 1PROOF. By construction in Notation 5.1.9, B :::; Na(D). Part (1) holds by
assumption, and says D acts on MK := Mn K = (Sn K)B. Thus D acts on
DB n (Sn K)B = B, completing the proof of (2). Further as the Borel subgroup
MK is 2-closed by 5.1.10, D acts on Sn K. As D acts on Sn Kand there are at
most two rank one parabolics of Kover Sn K, D acts on each such parabolic. So
as L 2 = P^00 for one of these parabolics, D acts on L 2 and hence also on Sn L 2.
By definition of S, S = 02 (BT), so B acts on S. As NL(S n L) =(Sn L)D,
DS =SD, completing the proof of (3). As B acts on SD, DSB is a group. By
5.2.1, S acts on L 2 , while by definition Bis a Cartan subgroup acting on L2. This
completes the proof of (4). DWe now verify that Hypothesis F.1.1 is satisfied with L, L 2 , S in the roles of
"L1", "L2", "S". Set B2 := BnL2, G1 := LSB2, G2 := DSL2, and G1,2 := G1nG2.
As L ::::l Mand B2 normalizes S by 5.2.4.3, G 1 is a subgroup of G with L ::::l G 1.
Again using 5.2.4, G2 is a subgroup of G with L2 ::::l G2. Thus Li= Gi as DSB
is solvable. Notice conditions (a), (b), and (c) of F.1.1 follow from remarks at thebeginning of the section, together with the fact that S acts on L 2. Further condition
( d) of F.1.1 holds as NLj (Sn Lj) :::; DSB :::; Gi, and we saw Li ::::l Qi. Condition
(f) follows from 1.1.4.5, since G1 :::; M, G2 :::; Na(K), and S contains 02 (M) and
02(H), and hence contains 02 (Na(K)) using A.1.6. Finally we establish (e) of
F .1.1 in the following lemma:LEMMA 5.2.5. 02((G1,G2)) = 1.PROOF. Let Go:= (G1, G2). By 4.3.2, M = !M(L), so as L 2 t;_ M, 02 (G 0 ) =
- D
We now use the Green Book [DGS85] (via an appeal to F.1.12) to deter-mine the possible amalgams that can arise; these will subsequently lead us to the
"generic" quasithin groups in conclusion (3) of Theorem 5.2.3, and to M 23 in con-
clusion (2) of 5.2.3.PROPOSITION 5.2.6. a:= (G1, G1, 2 , G2) is a weak EN-pair of rank 2. Further
L2 = K = G'f, with 02(Gi) = 02(Li) for i = 1 and 2, and one of the following
holds:
(1) a is the L3(2n)-amalgam and L and K are L 2 (2n)-blocks.(2) a is the Sp 4 (2n)-amalgam and L and K are L 2 (2n)-blocks.
(3) a is the G2(q)-amalgam for q = 2n, L/02(L) ~ K/0 2 (K) ~ L2(q),
02(K) ~ q^1 +4, and J02(L)J = q^5.
(4) a is the^3 D4(q)-amalgam for q = 2n, L/0 2 (L) ~ L 2 (q), J0 2 (L)J = q^11 ,
K/02(K) ~ L2(q^3 ), and 02 (K) ~ q1+^8 •
(5) a is the^2 F4(q)-amalgam for q = 2n, L/0 2 (L) ~ L 2 (q), J0 2 (L)J = q^11 ,
K/02(K) ~ Sz(q), and J0 2 (K)J = q^10.
(6) n > 2 is even, a is the U4(q)-amalgam for q = 2n/^2 or its extension of
degree 2, Lis an 04_(q)-block, K/0 2 (K) ~ L 2 (q), and 02 (K) ~ q^1 +4.
(1) n = 4, a is the U 5 (4)-amalgam, L/0 2 (L) ~ L 2 (16), J0 2 (L)J = 216 ,
K/02(K) ~ SU3(4), and 02 (K) ~ 41 +^6.