2.5. ELIMINATING THE SHADOWS WITH r 0 EMPTY
(i) K+ = KiK{, Ki -=f. Kl, Ki/021,2(Ki) ~ K, and K = CK+(t)^00 , or
(ii) K+ =Ki= [Ki, t] and K is a component of CKi (t).
561Set Ro := CR(K+)· In case (ii) as Ki/O(Ki) is quasisimple, 02 (Ki) ::::; Z(Ki),
so as m2(Ki) 2: m2(K) > 1, Ki is quasisimple by 1.2.1.5. Similarly if (i) holds,
then O(Ki) = 1 by 1.2.1.3, so that Ki is quasisimple. Thus in any case Ki is a
component of Gi.Let g E G with Ti:= Cru(i) E Syl2(Gi); then applying 1.1.6 to the 2-local Gi,
the hypotheses of 1.1.5 are satisfied with Gi, M9, z9 in the roles of "H, M, z".
Therefore Ki is described in 1.1.5.3. ·
Suppose for the moment that case (i) holds. Then by 1.2.1.3 applied to Ki, K
is not A5, so by 2.5.16, K is L 2 (p) for p 2: 7 a Fermat or Mersenne prime. Then
as Ki/Z(Ki) ~Kin (i), Ki ~ K by 1.1.5.3. Therefore all involutions in tK+ are
conjugate, and hence tzK E t^0 , so z = ZK by 2.5.13.3 and hence tz E t^0. Therefore
conclusion (I) of (3) and the first alternative in (4) hold in case (i). Thus in case
(i), it remains only to verify (1) and (2). Observe also in this case that NR(Ki)
centralizes the full diagonal subgroup K of K+, so Ro= NR(Ki) and R = (t) x R 0.
Next suppose for the moment that case (ii) holds. Comparing the groups in
2.5.16 to the components of centralizers of involutions in Aut(Ki/Z(Ki)) for groups
Ki on the list of 1.1.5.3, we conclude that one of the following holds:
(a) K ~ £ 3 (2), and t induces a field automorphism on Ki/Z(Ki) ~ £ 3 (4).
((3) K ~ £3(2), and t induces an outer automorphism on Ki/Z(Ki) ~ Jz.
('y) K ~ A 6 , and t induces one of: an inner automorphism on Ki/Z(Ki) ~HS,
an outer automorphism on Ki~ £4(2) or £ 5 (2), or a field automorphism on K; ~
Sp4(4).
Thus to prove that conclusion (II) of (3) holds in case (ii), it remains to show that
IZ(Ki)I ::::; 2 if (a) holds, and to show that Z(Ki) = 1 when ((3) holds, or when
Ki/Z(Ki) ~HS and ('y) holds.
Notice also when (ii) holds that from the structure of CAut(Ki/Z(Ki))(t) for the
groups in (a)-('y), either Ro= CR(Ki) is of index 2 in R, or else Ki/Z(Ki) ~HS
and in the latter case some r E R induces an outer automorphism on Ki, with
IR: Roi= 4, and CKi (R1)^00 ~As for some subgroup Ri of index 2 in R.
In the next few paragraphs, we will reduce the proof of 2.5.19 to the proof
of (2). So until that reduction is complete, suppose that (2) holds; that is that
R = (i, t) ~ E4.
We first deduce (1) from (2), so suppose that (1) fails. Thus Ko = KKu for
some u E S - N s ( K). Therefore i also acts on Ku, and hence also on S n Ku, so
that IC(i)(SnK")(i)I > 2. Since s n Ku ::::; Cs(K) and t tJ_ (i)(S n Ku) because t
centralizes Ko, IRI > 4, contrary to assumption. This contradiction shows that (2)
implies (1).
As remarked earlier, (1) and (2) suffice to prove the entire result when case (i)
holds. Thus to complete the proof of the sufficiency of (2), we may now assume
that case (ii) holds, and it remains to establish (3) and (4). Recall that at the start
of the proof we chose Cs((i,Ko)) E Syl2(Cc((i,t,Ko))), so as K =Ko by (1),
RE Syl2(Cc((t,i,K))).
As K :::] H, Ns(Cs(i)) acts on Cs(i) n Cs(K) = R. We saw i ¢_ Z(S), so
Cs(i) < Ns(Cs(i)). Then as Ns(Cs(i)) acts on R = (i,t), it E iNs(Cs(i))_ But
by A.3.18, Ki = 031 (E(Gi)), so i tJ_ t^0 by 2.5.18, and hence as it E i^0 , also
it tj_ t^0. As K::::; Ki= [Ki, t] and RE Sylz(Cc((t,i,K/)), (i/ = Co 2 (KiCs(i))(t).