526 2. CLASSIFYING THE GROUPS WITH IM(T)I = iSimilarly if L is a component of H, then by 1.1.5.3, L = [L, z] 1:. M, and the
possibilities for Lare listed in 1.1.5.3. Notice that Lis a component of (L, S) and S
is Sylow in N := Na(Q) by (1), so by 1.2.4, Ls LQ E C(N). Since L = [L, z], also
LQ = [LQ, z]. As N E I'o by (1), z inverts O(N) by (5); so as LQ = [LQ, z], LQ
centralizes O(N). Similarly z centralizes 02 (N) as z E Z(S) and SE Syl2(N), soLQ = [LQ, z] centralizes 02 (N). Thus LQ centralizes F(LQ), so LQ is quasisimple
by A.3.3.1, and hence LQ is a component of N. This completes the proof of (6).
To prove (7), we must refine the possibilities listed in 1.1.5.3. If L/Z(L) is aBender group, then Z(L) = 1by1.1.5.3, so conclusion (a) of (7) holds in this case.
Hence we may assume L/Z(L) is not a Bender group.
In this paragraph, we make a slight digression, to construct some machinery to
deal with groups of Lie rank at least 2. Assume L s (Hi, H2) with Hi E 1-le(S).
Suppose Hi 1:. M for some i. Then from the definitions in Notation 2.3.4, (S, Hi) E
U(Hi), so Hi E I' 0 by 2.3.7.3. Consequently Hi is described in 2.3.8.4.
Now suppose L/Z(L) appears in one of cases (a)-(c) of 1.1.5.3; then as L/Z(L)
is not a Bender group, L/Z(L) is a group of Lie type and characteristic 2 of rank
at least 2 in Theorem C (A.2.3). If there do not exist two distinct maximal Ns(L)-
invariant parabolics Ki and K 2 , then (cf. E.2.2.2) L/Z(L) ~ L 3 (2n) or Sp 4 (2n)'
with S nontrivial on the Dynkin diagram of L/Z(L), and then conclusion (b) of
(7) holds. Thus we may assume Ki and K 2 exist, take Hi := (Ki, S), and apply
the observations in the previous paragraph. By (6), L 1:. M, and hence Hi f; M
for some i, so Ki is a block described in 2.3.8.4. Then we check that the only
groups in (a)-(c) of 1.1.5.3 with such a block are those in conclusions (b) and (g) of
(7), keeping in mind that Z(L) = 02 (L) in case (b) of 1.1.5.3. Similar arguments,using generation by a pair of members of 1-le(S) in LS, eliminate those cases where
L/Z(L) is Mi 2 , M24, J 2 , J4, HS, He, or Ru; thus in case (f) of 1.1.5, L/Z(L) is
M11, M22 or M23·
If case (d) of 1.1.5.3 holds, then z has cycle structure 23 and as Ca(z) s M,
Ln M contains the stabilizer Kin L(z) of a partition of type 23 , 1 determined by z.
So as K is a maximal subgroup of L(z) and L 1:. M, K = MnL(z); thus conclusion
( c) of (7) holds.
In the cases L3(3), L2(p), Mu, M22, M 23 remaining from (e) and (f) of 1.1.5.3,the description of z determines the maximal subgroup of L(z) described in conclu-
sions (d), (e), (d), (f), and (f) of (7), respectively. Finally by 1.1.5.3, z induces aninner automorphism on L, except possibly when L/Z(L) is A 6 or A 7 , completing
the proof of (7).
Assume the hypotheses of (8). Because we are assuming that Q s R 2 Cs(R),
Co 2 (M)(R) S Cs(R) S R by (2). Then since IS : RI = 2 and J(S) S R byhypothesis, we have the hypotheses of 2.3.8.5b, and that lemma completes the
proof of (8), and hence of 2.3.9. DLEMMA 2.3.10. If S is of index 2 in T and 1-l(S) ~ M, then S E j3.
PROOF. As S ST s M, condition (/3 0 ) from the definition in Notation 2.3.1
holds. As IT : SI = 2, Na(S) S M = !M(T), and then the only proper 2-
overgroups of Sare Sylow groups T' of M, so (/3i) holds as M = !M(T'). Finally
by hypot~esis, there is HE 1-l(S) with H 1:. M; enlarging H if necessary, we may
assume H = Na(02(H)). As M = !M(T') for T' E Syb(M), SE Syb(H n M).