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s.

2. CLASSIFYING THE GROUPS WITH IM(T)I = 1

(b) If J(S) s Rs S with IS: RI= 2 and Co 2 (M)(R) SR, then RE (3.

(c) If H E 1-ie then Co 2 (M) (Ro) S Ro for each overgroup Ro of 02(H) in

(6) If H E r 0 , then the hypotheses of 1.1.5 are satisfied for each involution

z E Z(S) which is 2-central in M.

PROOF. As U E (3, S E (3 by 2.3.2.1, so S E Syl2(H n M). As S S M, we

may assume S s T. As M = !M(T), ISi < ITI completing the proof of (1). Part

(2) follows from 2.3.2.3. Next Na(0 2 (H)) E 1-i(H) ~rand (U,Hu) E U(H) ~

U(Nc(02(H))), so (3) holds.

Assume H E 1-ie n I'o. Then S E (3 by (1), and H E 1-ie(S, M) so that
(S, H) E U(H) and SEU. Assume that C(H, S) i M. Then there is a nontrivial
·characteristic subgroup R of S such that NH(R) i M. Now NH(R) E 1-ie using
1.1.3.2, so (S, NH(R)) E U(Nc(R)) and thus N 0 (R) E r. Then we may apply

2.3.7.3 with Na(R) in the role of "H 1 " to conclude that S E Syl2(Nc(R)). But

S < T by (1), so S < Nr(S) s Nr(R), contradicting S E Syh(Nc(R)). This

contradiction shows that C(H, S) s H n M. Then as SE Syl2(H), we may apply

the Local C(G, T)-Theorem C.1.29 to complete the proof of (4).

We next turn to (5), so we assume H E r 0 , and set J := J(S). By (1),

S E (3, so Z(T) s Co 2 (M)(S) s S using (f32) from the definition in 2.3.1. Then

D1(Z(T)) s D1(Z(S)) s J using B.2.3.7, so that J E Si(G) by 1.1.4.3. Suppose

Na(J) i M. Then (S, Na(J)) E U(Nc(J)) so Na(J) Er and SE Syh(Nc(J))

by 2.3.7.3. This is impossible as S < Nr(S) s Nr(J). Therefore Nc(J) s M,

proving part (a) of (5).

Next assume that HE 1-ie, and consider any Ro with Q := 02 (H) S Ro SS.

By 2.3.7.2, Nc(Q) E 1-i(H) ~ r 0 , with S Sylow in N 0 (Q) and NM(Q). Therefore

E := Co 2 (M)(Q) S 02(NM(Q)) SSS H.

Also F*(H) = 02(H) = Q as HE 1-ie, so Es CH(Q) SQ. Then as Q S R 0 ,

Co 2 (M)(Ro) S Co 2 (M)(Q) =ESQ S Ro,

establishing part ( c) of ( 5).

So to complete the proof of ( 5), it remains to establish part (b). Thus we

assume that J s R s S with IS : RI = 2, and Co 2 (M) (R) s R. We must show

that R E (3; as R satisfies (f3o) since S s M, and R satisfies ((3 2 ) by hypothesis, we

may assume that ((3 1 ) fails for R, and it remains to derive a contradiction. Then

for some Rs VE S2(M), Na(V) i M, and we may choose V maximal subject to

this constraint. As usual, we may assume that V s T. By hypothesis J(S) s R,

so J(S) = J(R) by B.2.3.3, and hence Na(R) s N 0 (J(S)) s M by part (a) of

(5). Therefore R < V. Further Na(V) i M = !M(T), so that V < T and hence

V < Nr(V) := W. Then W satisfies ((3 0 ), and also ((3 2 ), since this condition is

inherited by overgroups of R. Further by maximality of V, Na(X) s M for each X

satisfying W s XE S2(M), establishing ((31) for W. Hence WE (3. We saw earlier

that J(S) = J E Si(G), and by hypothesis J s Rs V, so V E Si(G) by 1.1.4.1,

and hence Na(V) E 1-ie. Then (W,Na(V)) E U(Nc(V)) so Na(V) Er. However

by hypothesis IS: RI= 2, while R < V < W, so that IWI > ISi. This contradicts

the maximality of IHl2 in Notation 2.3.5 when HE r*, and the maximality of IUI

when HE r*. This contradiction completes the proof of (5).