7.7. MINI-APPENDIX: r > 2 FOR L 3 (2).2 ON 3 E9 3 709We now eliminate the cases (a)-(d), (e) with K* ~ A 6 , and (f); in all these
cases, K is a block. We have V = [V, £^0 ] :::; K using 7.7.10. Recalling that
V :::; U :::; 02 ( H), we see that V _ :::; 02 ( K). Let W be the unique noncentral 2-chief
factor of the block K, and W the image of V in W. As Cv (£^0 ) = 1, W ·~ V_.
Further Q centralizes W _ and Q is of index 2 in the Sylow group To. However in
each case, W is of dimension 4 or 6, and no subgroup of index 2 in a Sylow groupcentralizes a 4-subspace of W.
We are left with case (h), and with the subcase of case (g) where n = 1. Thus
K* ~ Lm(2) with m := 3, 4, 5. As £^0 * is normal in the parabolic Mk and T 0 -invariant, L^0 *TK: is a rank one parabolic determined by a node t5 in the Dynkin
diagram adjacent to no node in MK- So when mis 4 or 5, unless K*T 0 ~ 88 and
t5 is the middle node, there is an £^0 T 0 -invariant proper parabolic which does not
lie in M, contrary to the minimality of H. When K*T 0 ~ 88 , Theorems B.5.1 and
B.4.2 say I:= [U, Kl/C[u,K] (K) is either the orthogonal module or the sum of thenatural module and its dual. But in either case, m(C1(To)) = 1, impossible as V_
is isomorphic to an £^0 T 0 -submodule of I and m( Cv_ (To)) = 2.
Therefore K* ~ £ 3 (2), and C.2.7.3 says that K is described in Theorem C.1.34.
As m(Cv_(To)) = 2, there are at least two composition factors on U:::; Z(0 2 (K)),
ruling out all but case (2) of C.1.34. Hence 02 (K) = U = U 1 EB U 2 is the sum
of two isomorphic natural modules for K* = °K/U, with V_ = W 1 EB W 2 where
Wi =Cui (Q). Then an element e of £^0 of order 3 has a unique nontrivial composi-
tion factor on 02 (£^0 *), (which is realized on Q/U) plus two nontrivial composition
factors W 1 and W 2 in U (realized in V). Thus £^0 has just one nontrivial com-
position factor on Q/V, which is impossible since the outer automorphism f of
L ~ £ 3 (2) must interchange any·natural module and its dual, and these are the
only irreducibles with a unique nontrivial £^0 -composition factor. This contradic-
tion finallY. completes the proof of Proposition 7.7.3 and hence also of Theorem
7.7.1.