868 i3. MID-SIZE GROUPS OVER F2PROOF. Assume XE B'f_(G,T). Then XE B*(G,T), so Mc:= Nc(X) =
!M(XT) by 1.3.7. Also X E B+(G, T), so by 3.2.13, X (j!:_ Bt(G, T). Then by
A.4.11, X centralizes R 2 (XT), so R 2 (XT) contains Z by B.2.14; hence Mc =
!M(XT) = !M(Cc(Z)), so that (1) holds. This also establishes (2), as we may
vary XE B'f-(G, T) independently of Z. D
LEMMA 13.1.6. Assume X E B'f_(G, T), let Mc E M(XT), and assume ME
M(T) - {Mc}· Then either
(1) There exists an odd prime d and Y = 02 (Y) ::::l M such that Y i. Mc,
[Z, Y] -:f=. 1, and Y/0 2 (Y) is ad-group of exponent d and class at most 2, or
(2) There exists Y E C(M) with Y i. Mc. For each such Y, Y/02(Y) is
quasisimple, Y ::::l M, [Z, Y] -:f=. 1, and YE £j(G,T).PROOF. By 13.1.5, Nc(X) = !M(XT) =Mc= !M(Cc(Z)).
Suppose first that there is YE C(M) with Yi. Mc. Then as Mc= !M(Cc(Z)),
[Z, Y] # 1, so that Y E Lt(G, T). Let Y ::::,:; Yi E £*(G, T); by 1.2.9.2, Yi E
£j(G, T). If Yi E £+(G, T), then by (2) and (6) of 13.1.4, Nc(Yi) = !M(YiT) =
!M(Cc(Z)), contrary to our assumption that Y i. Mc. Thus Yi/02(Yi) is qua-
sisimple, so that Y =Yi ::::l M by 13.1.2. Therefore (2) holds in this case.
We may assume that (2) fails, so (C(M)) :::::; Mc by the previous paragraph. Let
M* := M/02(M), and ford an odd prime, let ed(M) be the preimage of the group
ed(M) defined in G.8.9; recall that ed(M) is of class at most 2 and of exponent
d using A.l.24. Let B(M) be the product of the groups ed(M), ford E n(F(M*)).
Suppose that B(M):::::; Mc. Then as (C(M)):::::; Mc, B(M)02,E(M) =: Y::::,:; Mc.
with M, Mc in the roles of "H, K", R := 02 (Mc n M) = 02(M) and C(Mc, R) :::::;
Mc n M. Then Mc, R, Mc n M satisfy Hypothesis C.2.3 in the roles of "H, R,
MH". As X E B+ ( G, T), X is a {2, p }-group for some prime p > 3, so X contains
no A3-blocks. Thus by C.2.6.2, X:::::; Mc n M, contrary to M #Mc= !M(XT).
This contradiction shows that B(M) i. Mc; hence there is some d with Y :=
ed(M) i. Cc(Z) and Y = 02 (Y) ::::1 M; so (1) holds. D
We are now prepared for the main result of the section:THEOREM 13.1.7. Assume Hypothesis 13.1.1. Then £+(G, T) = 0.
Until the proof of Theorem 13.l.7 is complete, assume G is a counterexample.
As L+(G,T) is nonempty, we may choose LE £'f_(G,T), so LE £j(G,T) by an
earlier remark. Set Mc := Nc(L); then Mc= !M(LT) by 13.1.4.2. By Theorem
2.1.1, IM(T)I > 1, so 1-l*(T, Mc) is nonempty.
Let X consist of the groups Bp(L), p E n(F(L/02(L))). By 13.1.4, each XE X
is in B'f-(G, T) and
Mc= Nc(X) = !M(XT) = !M(Cc(Z)).
Set Vc := R2(Mc), M~ := Mc/CMJVc), and
U := [Vc,L].
Define
£i :={Li E £(G, T) : L = Ooo(L)Li}.
LEMMA 13.1.8. (1) L* ~ L2(P) for some prime p > 3.
(2) 1 -:f=. [Ve, L] =[Ve, Li] for each Li E £i.(+)