7.4. IMPROVED LOWER BOUNDS FOR r 699In view of 7.3.2, the column headed m ;?: in Table 7.2.l also provides a lower
bound for the parameter s. Then comparison with a gives us information on w.Recall from Definition E.3.23 that
w := min{m(V^9 /V^9 n T): g E G and [V, VB n T]-/= l}.
LEMMA 7.3.3. The column ''w;?:" of Table 7.2.1 gives a lower bound for w.
PROOF. Recall from E.3.34.l that w;?: s - a. Ass= m by 7.3.2, we subtract
the column for a from the column for m in the Table, and obtain the result. DHaving established a lower bound on w, we now apply E.3.35 in order to obtain
an upper bound for w.
Let H denote an arbitrary member of 'H*(T, M), although from time to timewe may temporarily impose further constraints on H.
PROPOSITION 7.3.4. w :S n(H) :S n'(Mv) = n' < s, where n' is listed in the
column headed ''n' " in Table 7. 2.1.
PROOF. Let k denote the value of n' given in Table 7.2.1; we first assume
n' = k. Recall that s = m by 7.3.2, and observe further that m > n' in all cases
in the Table, so that s > n'. Next we check that Hypothesis E.3.36 is satisfied:
We observed in the introduction to this chapter that V :::! T, M = !M(Na(Q)),
and Vis a TI-set under M, with H :S Ca(Z), and H n M :S CM(Z) :S NM(V).
Further by Hypothesis 7.0.2, Vis neither an FF-module nor the orthogonal module
for L 2 (2^2 n), so whenever n(H) > 1 we can apply Theorem 4.4.14 to conclude that
a Hall 2'-subgroup B of H n Mis faithful on L 0 , and hence also on V. It follows
that CHnM(V) :S 02(H n M), completing the verification of Hypothesis E.3.36.
Now since n' < s :Sr, the lemma holds by E.3.39.1.
Thus it remains to verify that k = n^1. If Lis £ 3 (2) on 3EB3 or Sp 4 (2)' ~ A 6 on
4 EB 4, then T is nontrivial on the Dynkin diagram of L, and hence T permutes with
no nontrival subgroup of Mv of odd order, so that n' = 1 = k. In all other cases
where L is of Lie type, T permutes with a Cartan subgroup of L, which contains a
cyclic subgroup of order 2k - 1, so that n' ;?: k in these cases. Similarly when L is
sporadic, T permutes with a subgroup of order 3 and k = 2, so n' ;?: k. Finally if
n' > k then n' > 2 and we may apply A.3.15 to some prime p > 3 which does not
divide k(2k - 1) and obtain a contradiction which completes the proof. D
We can already see that when Lis Sp 4 (2n), the value in the column w;?: strictly
exceeds the value in the column n', so that 7.3.3 and 7.3.4 provide our first example
of a numerical contradiction, eliminating one of our cases from Table 7.1.1:
COROLLARY 7.3.5. L is not Sp 4 (2n)'.^2
7.4. Improved lower bounds for r
We saw earlier in 7.3.2 that r ;?: m ;?: 2. In many cases, we can improve this
bound on r using E.6.28: First r > 1, giving hypothesis (1) of E.6.28. As V is
not an FF-module, hypothesis (2) of E.6.28 holds. Finally if L :::! M, and X is
an abelian subgroup of CM(V) of odd order, then Na(X) :SM by Theorem 4.4.3.
(^2) It would also be possible to eliminate case (iii) of 3.2.6.3.c at this point (adjusting for the
fact that V might not be a TI-set under M). However, it seems more natural to treat all cases of
3.2.6.3.c uniformly in chapter 10.