2.5. ELIMINATING THE SHADOWS WITH r(; EMPTY 563(A) If R < Cs(K) choose 8 E Cs(K).
(B) If R = Cs(K), choose 8 E Ns(K); we check this choice is possible: When
K =Ko this is trivial, while when K < K 0 , by assumption i ~ Z(Ns(K)), so again
the choice is possible.In either (A) or (B), 8 E Ns(K). Hence as 8 E Ns(Cs(i)), 8 normalizes
Cs(i) n Cs(K) = R.
In case (A) set W := R(8), and in case (B) set W := Cs(i)(8). In either case,
W = Cw(i)(8). Furthermore 82 E Cw(i): As 82 E Cs(i), this is immediate fromthe definition of W in case (B), while in case (A) we chose 8 E Cs(K), so that
82 E Cs(i) n Cs(K) = R = Cw(i).
We now show that R2 :s1 Cw(i): In case (A), this holds as R 2 is of index
2 in R = Cw(i), so assume case (B) holds. Then Cw(i) = Cs(i) normalizes
Cs(i) n Cs(K+) =Ro, so the claim holds when IR : Roi = 2, since in that case
R2 =Ro. Thus we may assume IR: Roi= 4 and Ki/Z(Ki) ~HS, so that R 2 isthe subgroup Ri of Ro with a component A 8 in its centralizer. But Cs(i) acts on
the 4-group R/ Ro, and hence also on the unique subgroup Ri/ Ro of order 2 with
K < E(CKi(R1)). So indeed R2 :s1 Cw(i).
As R 2 :s1 Cw(i) and 82 E Cw(i), W = Cw(i)(8) normalizes R 2 n R?,. Assume
R2 n R?, f 1; then CR 2 nR2(W) f 1. Let r be an involution in CR 2 nR2(W); fromthe definition of R 2 , K is not a component of Ca(r). In case (A), R < W :::;
C 8 (r) n C 8 (K), contrary to the maximality of R. In (B), R = C 8 (K) :::; Cs(i) <
W :::; Cs(r), contrary to the maximality of C 8 (i) in our choice of i, R under the
constraint that R = Cs(K). Therefore R2 n R?, = 1, so as IR: R21 = 2, IRI = 4,
contrary to our assumption that Rf (i, t). This finally completes the proof of the
claim.
By the claim, Ko = KKu for u ES - Ns(K) and i E Z(Ns(K)). Thereforeby 1.2.i.3, K is described in case (1) or (2) of 2.5.16, so K ~ L2(P) for p ~ 7 a
Fermat or Mersenne prime. In case (i) we showed that Ki~ K, so K+ is the direct
product of two t-conjugates of a copy of K. In case (ii), K ~ Ls(2), so (a) or ((3)
holds.
Let j be an involution in Ro= CR(K+), Gj := Ca(j), Lo:= (K~^2 ',E(Gj\ and
L+ := (K~^2 ',E(Gj\ Then K < K+ :::; Gj, so as K is not subnormal in K+, K isnot a component of Gj. Indeed we claim that K+ :s1 Gj. As Ki is a component
of Cai (i), we may apply the initial arguments of the proof of 2.5.19 to j, i, Ki in
the roles of "i, t, K". We conclude that there is a component L of Gj such that
either L =Lo is i-invariant, or L <Lo= LLi with CLo (i)^00 a component of Cai (i)
isomorphic to Ki ~ L 2 (p) for suitable p. It follows that L+ = LoLg. Similarly in
case (i) where K ~Ki, if L =Lo we may apply 1.1.5 to conclude that Lis L 3 (4)
or J2 of 3-rank 2.
If K+ = L+, then we conclude from A.3.18 in case (ii) or from 1.2.2 in case (i)that L+ = 031 (E(Gj)) :s1 Gj. Thus to establish the claim that K+ :'9 Gj, it will
suffice to show that K+ = L+.
Suppose that case (ii) holds. Then Ki is described in (a) or ((3), so that 1.2.1.3.
rules out the case L <Lo. Thus Lo= L, and L = [L, t] as tacts on Ki· Then our
earlier argument applied tot, j, Kin the roles of "t, i, K" shows that Lis L 3 (4)
or J 2. But then as Ki is a component of CL(i), L =Ki. Then as L =Lo= Ki,
L+ = LLt = KiKf = K+, as desired.