2.3. PRELIMINARY ANALYSIS OF THE SET ro 525It remains to prove (6). By (1), S :::; M and S E Syb(H n M). Assume
that H E I'o. Then by 2.3.7.2, Na(02(H)) E r, and Sis Sylow in Na(0 2 (H))and NM(02(H)). Thus Co 2 (M)(02(H)) :S: 02(NM(02(H))) :::; S. Now 02 (H) :::;
02(H n M) by A.1.6, so
Co2(M)(02(H n M)):::; Co2(M)(02(H)):::; s:::; H,
establishing one of the hypotheses of 1.1.5. Finally if z is an involution central inT' E Syb(M), then Ca(z) ::::; M = !M(T'), establishing the remaining hypothesis
for that result. This establishes (6), and so completes the proof of 2.3.8. D
The final section of this chapter will focus on components of a member of r 0 •
Using part (6) of 2.3.8, the next result describes these components.LEMMA 2.3.9. Let HE I'o, Q := 02(H), (U,Hu) E U(H), and U:::; SE
Syb(H). Then
(1) S is Sylow in Na(Q) and NM(Q), and Na(Q) E I'o. If H E I'*, then
Na(Q) EI'*.(2) Co 2 (M)(Q) :S: S.
(3) Z(T) ::::; S < T for some T E Syb(M) depending on H. In particular,
Z(T)::::; Z(S).
(4) F*(H n M) = 02(H n M).
(5) Let z be an involution in Z(T) for T as in (3). Then Ca(z) ::::; M and z
inverts O(H).
(6) If L is a component of H, then L = [L, z] 1:. M, and L is contained in acomponent LQ of Na(Q).
(7) If L is a component of H then z induces an inner automorphism on L
unless possibly L/Z(L) ~ A 6 or A1. Moreover one of the following holds:
23, 1.
z on L.
(a) L is a Bender group.(b) L ~ Sp4(2n)' or L3(2n), or L/02(L) ~ L3(4) or L ~ A5.
(c) L ~ A 7 or A1, and L n M is the stabilizer in L of a partition of type
(d) L ~ £ 3 (3) or Mu, and L n M = CL(ZL) where ZL is the projection of
(e) L ~ L 2 (p), pa Fermat or Mersenne prime, and L n M =Sn L.
(f) L ~ M22 or M23, and L n M ~ A5/ Em or A1 / E15, respectively.(g) L ~ £ 4 (2), S is nontrivial on the Dynkin diagram of L, and L n M =
CL(zL), where ZL is the projection of z on L.
(8) Assume IS : RI = 2, with R containing J(S), 02(H), and Cs(R). Then
RE (3.
PROOF. By 2.3.8.3, Na(Q) E I'; then (1) follows from parts (2) and (4) of
2.3.7.
By (1), Sis Sylow in NM(Q), so Co 2 (M)(Q) :::; 02(NM(Q)) :::; S, proving (2).
By 2.3.8.l, S E (3, so in particular S ::::; M and S < T for some T E Syb(M), AsF*(M) = 02 (M), Z(T) :::; 02 (M), so as Q:::; S:::; T, Z(T) :::; S by (2), completing
the proof of (3).
By (3), Z(T) ::::; Z(S), so by 2.3.8.6 the hypotheses of 1.1.5 are satisfied for each
involution z E Z(T), and in particular Ca(z) ::::; M. Therefore 1.1.5.1 implies (4),
while 1.1.5.2 says z inverts O(H), completing the proof of (5).