16.5. IDENTIFYING J1, AND OBTAINING THE FINAL CONTRADICTION 1203(b) L ~^2 F4(2n)', X/02(X) ~ L2(2n)', Z(02(X)) = V EB W, where W :=
[Z(0 2 (X),X] is the natural module for X/0 2 (X), and V = Z(X).
(c) L ~^3 D4(2n), X/02(X) ~ L2(2n)', Z(02(X)) = V EB W, where W :=
[Z(0 2 (X)),X] is the natural module for X/02(X), and V = Z(X) ~ E 2 n.
In case (a), set W := 02 (X). We finish as in several earlier arguments: In each
case, W# is the set oflong root involutions in Z(0 2 (X)), and vv^1 E W# for suitable
l E £, contrary to 16.5.10.6.
This final contradiction establishes:
THEOREM 16.5.14 (Even Type Theorem). Assume G is a quasithin simple
group, all of whose proper subgroups are K-groups. Assume in addition that G is