lS.3. THE ELIMINATION OF Mr/CMr (V(Mr)) = 83 wr Z 2 1131By 15.3.26.2, L = [L, Y+J. Comparing the list of Theorem B.5.6 with that in
A.3.18, we conclude that one of the following holds:
(i) m3(L) = 1 and L* is L2(2n) or L 3 (2m), m odd.(ii) L = ()(I).
(iii) L* S:! SL3(2n), n even.
In case (ii), (2) holds. In case (i), as Y+S/R S:! S 3 and Out(L/0 2 (L)) is abeHan,
Y+ induces inner automorphisms on L *. Then the projection Y£ of Y+ on L * is
contained in Mj by 15.3.26.3, so the preimage YL is contained in 03 ' (Mr) = Y+
by 15.3.26.1, so that (2) holds again. Finally if (iii) holds, then Z(L)* _:::; Mj by
15.3.26.3, so 031 (Mr)* = Y+ = Z(L*) by 15.3.26.1, contradicting L* = [L*, Y+J.
This completes the proof of (2), and the same argument shows that L* is not A 6 ,
completing the proof of (3) also.
Finally assume that IE 'H+,*· By (1), S acts on L, and by (2), Y+:::;; L. Thus
as Li M, LS E 'H+, so I= LS by minimality of I. Similarly Mr is the unique
maximal subgroup of I containing Y+S, so (4) holds. DUntil the proof of Theorem 15.3.25 is complete, assume IE 'H+,*· Thus I= LS
by 15.3.27.4.LEMMA 15.3.28. Let ML :=.Mn L; then one of the following holds:
(1) L S:! L2(2n), n even, and MI, is a Borel subgroup of L.
(2) L S:! L3(2), Sp4(2)', or G2(2)', and Ml, is a minimal parabolic of L, so
that S is trivial on the Dynkin diagram of L *.
(3) I S:! Ss and MJ, is the middle-node minimal parabolic of L S*.
PROOF. Suppose first that L S:! A 7. Then as Y+S/R S:! S 3 by 15.3.26.1, Y+S
is either the stabilizer of a partition of type 23 , 1, or is contained in the stabilizer
14, 3 of a partition of type 4, 3. In the latter case, the preimage 14 , 3 is contained in
Mr by 15.3.27.4, whereas m3(Mr) = 1by15.3.26.1. In the former, Y+S* :::;; Ii S:! A6
or S 6 , and this time Ii :::;; Mr for the same contradiction.
Then by 15.3.27.3 and Theorem B.5.6, L* is of Lie type and characteristic 2,. so as S E Syb(I), MI, is a maximal S-invariant parabolic of L* by 15.3.27.4. As
03 ' (ML)= Y+ with Y+S/R S:! S 3 by 15.3.26.1, we conclud~ by inspection of the
list of Theorem B.5.6 and appeals to parts.(3) and (4) of 15.3.27 that one of cases
(1)-(3) of the lemma holds. DLEMMA 15.3.29. (1) No nontrivial characteristic subgroup of Sis normal in I.
(2) NG(V1) :SM and Vi centralizes Y+·PROOF. As Y+ :::;; L by 15.3.27.2 and I= LS, we may apply 15.3.11.12 to obtain
(1). By 15.3.26.1, case (2) of Hypothesis 15.3.10 holds, so that Vi centralizes Y+,
and NG(V1) :S M by 15.3.11.3, so (2) holds. DLEMMA 15.3.30. L* is not L2(2n).. PROOF. Assume L is L 2 (2n). By 15.3.28, MI, is a Borel subgroup of L, so as
R = 02 (Y+S), RE Syb(LR). Then by 15.3.27.4, LR is a minimal parabolic in the
sense of Definition B.6.1, so we conclude from 15.3.29.1 and C.1.26 that Lis a block