8.2. DETERMINING LOCAL SUBGROUPS, AND IDENTIFYING J 4 715the hypotheses of 3.1.9 hold with LT in the role of "M 0 ": Recall we saw after
Hypothesis 7.0.2 that H n M::::; Mv; thus case (II) of Hypothesis 3.1.5 holds. Now
part (c) in the hypothesis of 3.1.9 holds by hypothesis, part (a) is a consequence of
Table B.4.5, and (d) follows as the dual V of V satisfies q(L'i', V) > 2. Finally
M = !M(LT) by 1.2.7.3, so (b) holds.
By A.3.18, L = 031 (M). Then we observe that each element of order 3 in M 22
has six 3-cycles on 22 points, so it has three noncentral chief factors on each of Vand V*. Set Lz := 02 (CL(z)); then Lz = 02 (CM(z)) and Lz/02(Lz)) ~ A 6. Thus
. each {2, 3}'-subgroup of CM(Z) permuting with T centralizes V. As q(Mv, V) = 2,
but m(Mv, V) = 3 by H.14.4, each member of Q(Mv, V) has rank at least 2. Thus
3.1.9.6 says that H/0 2 (H) ~ 83 wr Z 2 or D 8 /31+^2 ; in particular X := 02 (H) EB(G, T). Next by 1.2.4, Lz ::::; Kz E C(Ca(z)); and by A.3.12, either Kz = Lz
or Kz/02(Kz) ~ A1, Mn, M22, M23, or U3(5). By A.3.18, Kz = 0
31
(Ca(z)),so X ::::; Kz. Thus Kz/02(Kz) ~Mu by 1.3.4. But then H/02(H) ~ SD15/E9,
impossible as H/02,3(H) ~ D 8. This completes the proof of (1).
Then as H is a minimal parabolic, V 1. kerMnH(H) by B.6.8.5, so that Hy-
pothesis E.2.8 holds. Then (2) follows from E.2.9. By E.2.11.5, 02 (1) = kerMI(I).By 7.3.3 and 7.5.6, w > O, so Wo(T, V) centralizes V. Therefore I* is not Sp 4 (2k)
by E.2.13.5; in particular the remainder of (5) holds by definition of 'I(H, T, V).
As q(M, V) > 1, E.2.13.4 says that (3) holds. We recall from the introduction
to the previous chapter that V is a TI-set under M, so that with (3), we have
the hypotheses of E.2.14. Now (4) follows from the definition of 'I(H, T, V) and
E.2.14.1, while (6) follows from E.2.14.2. The first few statements in (7) follow
from E.2.13.1 and E.2.15. Then we compute m(A) using (5), (6), and the factthat CA(V) = Zr. By E.2.10.1, AB :S! I, while by parts (3), (4), or (10) of
E.2.14, Cr(AB) ::::; kerMAI) = 02(I); thus Cv(A) ::::; Cv(AB) ::::; 02(I) n V = B,
completing the proof of (7). Finally, (8) follows from (5) using E.2.14.9. DAs in 8.2.2, pick I= (V, Vh) E 'I(H, T, V)), and adopt the rest of the notation
established in the lemma; e.g., Tr:= TnI E Syl2(I), Mr:= MnJ, I*:= I/02(I) =I/kerMAI), k := n(I), etc.
PROPOSITION 8.2.3. k = n(I) = w = n, 1, 1, 2, and A is aw-offender on V.
PROOF. By 8.2.2.5, k divides n(H), so k ::::; n(H). By definition w ::::; m(V*),
while m(V*) = k using 8.2.2.6. Then we can extend the inequality in 7.3.4 tow::::; m(V) = n(I) = k::::; n(H)::::; n' = 2n,2,2,2 ()
using the values in Table 7.2.1.
In the fourth case M24/ll, w = 2 by 8.1.3, so the lemma follows from (*).
Thus we may assume Lis not M 24 on 11. If w = k, then A is aw-offender.
By Table 7.2.1 and 7.5.6, w 2 n, 1, 1. Thus if k ::::; n, 1, 1, then w = k by (*) and
the lemma holds. Therefore by (*),we may assume that k = 2 if Lis M 22 or M 2 4,
while n < k::::; 2n if L ~ (S)L 3 (2^2 n), and it remains to derive a contradiction.
Assume first that L ~ (S)L 3 (2^2 n). Then k > n 2 1, so I* ~ L 2 (2k) or
Sz(2k) and hence Autr(V) contains a cyclic subgroup X of order 2k - 1 2 3acting nontrivially on A. Therefore as Out(L) is 2-nilpotent, 1 #- [A, X] ::::; L is
an X-invariant 2-group. Hence X acts on some parabolic of L, and indeed on a
maximal parabolic as X has odd order. Therefore 2k - 1 divides (2^4 n - l)n, so as
n < k ::::; 2n, it follows that k = 2n. Thus m(A) ::::; m2 = 4n = 2k, so by E.2.14.7,