986 i4. L 3 (2) IN THE FSU, AND L 2 (2) WHEN .Cf(G, T) IS EMPTYof two conjugates of Ki, since case (2) of 14.2.8 holds~see e.g. 16.1.4 and 16.1.5.
This contradiction completes the treatment of the case that L is quasisimple.
· Therefore Go E He, so Vo := (ZG^0 ) E R 2 (G 0 ) by B.2.14. Then [Vo,L] -::/= 1
since we saw [L, Z] -::/= 1. If C is a nontrivial characteristic subgroup of 8 with
L:::; Na(C), then H =KT :S (L,T) :S Na(C), so L :S Na(C) :S Mc= IM(H)by 14.2.3, contradicting [L, Z] -::/= 1. Hence no such C exists, so as L/02,P(L) is
quasisimple by 1.2.1.4, L = [L, J(8)]. Then appealing to Thompson FactorizationB.2.15, V 0 is an FF-module for L8/CLS(V 0 ), so by Theorems B.5.1 and B.5.6,
L/CL(Vo) ~ L2(2n), 8L3(2n), 8p4(2n), G2(2n), Ln(2), A5, or A1. As Ki <Land
8 acts on Ki with n(Ki) > 1, L/CL(Vo) is not Ln(2) or a group over F2 or A5,
and also Lis not a xo-block. Further L/0 2 (L) is not A 7 , since the FF-modules in
Theorem B.5.1 do not satisfy the condition [K 1 , Zs]= 1 in (I). Therefore L/CL(Vo)
is 8L3(2n), Sp4(2n), or G2(2n), and Ki/02(Ki) ~ L2(2n) for n > 1. Recall
R = 02 (Y8). If Yi. L, then as we observed earlier, RE 8yl 2 (LR); while if Y :SL
then Y is contained in a Borel subgroup of L, and then once again, R is Sylow in
LR. We also saw C(G, R) :::; M, while Li.Mas K 1 i. M; thus Lis a x 0 -block byC.1.29, contrary to an earlier observation. This contradiction completes the proof
of 14.2.15. D
LEMMA 14.2.16. a is a weak EN-pair of rank 2, K = K 1 , T = 8, Q :=02(K) = 02 (Mc) is extraspecial, and either
(1) a is isomorphic to the^3 D 4 (2)-amalgam, [Qf = 2i+s, and K/Q ~ L 2 (8), or
(2) a is parabolic isomomorphic to the J 2 -amalgam or Aut(J 2 )-amalgam, [Q[ =
21 +4, and K/Q ~ L2(4).
PROOF. Recall 14.2.15 completed the verification of Hypothesis F.1.1 with
Ki, Y8 2 , 8 in the roles of "Li, L 2 , S". Then by F.1.9, a is a weak EN-pair
of rank 2. Furthermore we saw B 2 = 8 2 , so a appears in the list of F.1.12. SinceG2/Ca 2 (V) 3:: 83, while Ki is nonsolvable and centralizes Z, we conclude that
a is either isomorphic to the^3 D4(2)-amalgam, or is parabolic-isomorphic to the
h-amalgam or the Aut(J2)-amalgam. In each case Zs ~ Z 2 , (Z§) ~ E 4 , and
Q = 02 (Ki) = 02(Ki8) is extraspecial of order 2i+s or 2iH, while Ki/Q ~ L 2 (8)
or L2(4).
As Zs is of order 2, Z = Zs. Also K1 is irreducible on Q/Z, so Q = 02 (Mc)
using A.1.6. Further the action of Ki on Q/Z does not extend to (8)L 3 (2n),
8p 4 (2n), or L 2 (2n) x L 2 (2n), so as K = (K'{), case (1) of 14.2.8 holds, so K =Ki
and T = 8. · · DWe say G is of type J3 or J2 if a is parabolic isomorphic to the J 2 -amalgam,
and G has 1 or 2 classes of involutions, respectively.
LEMMA 14.2.17. Assume a is parabolic isomorphic to the Jramalgam or the
Aut(J 2 )-amalgam. Then(1) a is parabolic-isomorphic to the J2-amalgam, and G is of type J 2 or J 3.
(2) If G is of type J2, then G ~ J2.
(3) If G is of type J3, then G ~ J3.
PROOF. By 14.2.16, Q = 02(Mc), so as Out(Q) ~ 85, KT= Mc= Ca(Z).Assume first that a is parabolic isomorphic to the Aut(J 2 )-amalgam. Then by
46.1and46.11 in [Asc94], K has three orbits on involutions in K, with representa-
tives z E Z, s E V-Z, and t EK -Q with CT(t) E 8yl2(CKT(t)). Thens E zY, so