io84 i5. THE CASE £,f(G, T) =^0
PROOF. As Mis maximal in M(T) under :Sand V = V(M), we conclude from
A.5.7.2 that R = 02(NM(R)), V E R2(NM(R)), M = AutNM(R)(V) = NM(R),
and M = !M(NM(R)). So as V ::::] M, (1) holds. Further for H E 1-l*(T,M),
02 ((H,NM(R))) = 1 as M = !M(NM(R)), so conclusion (1) of Theorem 3.1.6
does not hold. Then conclusion (2) or (3) of 3.1.6 holds, establishing (2) since
NM(R) =M. D
By 15.1.1, AutM(V(M)) and its action on V(M) are described in section D.2.
Using the fact that Mc= !M(Ca(Z)), we refine that description in the first lemma
in this section, which provides the basic list of cases to be treated in the first three
sections of this chapter. Recall Q*(AutM(V(M)), V(M)) from Definition ·D.2.1,
and set }(AutM(V(M)), V(M)) := (Q*(AutM(V(M)), V(M))).
LEMMA 15.1.2. LetV := V(M), andsetM := M/CM(V) andMJ := }(M, V).
Then one of the following holds:
(1) MJ ~ D 2 p and m(V) = 2m, where (p, m) = (3, 1), (3, 2), or (5, 2).
(2) m(V) = 4 and M = MJ = nt(V) ~ 83 x 83.
(3) MJ = Mi x M2 and V = Vi EB Vi, with Mi ~ D2p, Vi := [V, Mi] of rank
2m, (p, m) as in (1), and Mi and M 2 interchanged in M.
(4) MJ = P(f'; where P := 02 (M) ~ 3i+2, and tis an involution inverting
P/iJJ(P). Further m(V) = 6, and Tacts. irreducibly on P/iJJ(P).
(5) MJ = P(f'; where P := 02 (M) ~ E 9 and tis an involution inverting P.
Further m(V) = 4, and f' ~ Z4.(6) MJ ~ 83, V = [V, MJ] x Cv(MJ) with m([V, MJ]) = 4 and Cv(MJ) #-1,
M/CM([V,MJ]) = nt([V,MJ]), and Mn Mc= CM([V,MJ])CM(Cv(MJ))T is of
index 3 in M.
PROOF. By 15.1.1.1, iJ.(M, V) ~ 2, while by 14.1.6.1, M is solvable. Hence, in
the language of the third subsection of section D.2, (MJ, [V,F(MJ)]) is a sum of
indecomposables, so there is· a partition
Q*(M, V) = Qiu··· u Qs
such that MJ = Mi x · · · x Ms and Vo := [V, F(MJ )] = Vi EB · • · EB Vs, where
Mi :=(Qi), Vi := [V, Mi], and (Mi, Vi) is indecomposable as defined in section D.2.
Further by D.2.17, each indecomposable (Mi, Vi) satisfies one of the conclusionsof D.2.17. Let Mi denote the preimage in M of Mi. As M permutes the set
{Qi: 1 ~ i ~ s} of orbits of MJ on Q*(M, V), M permutes {Mi: 1 ~ i ~ s}.
Observe that F*(Mi) = Op(M) for some odd prime p (depending on i), so for
each nontrivial 2-element tin Mi, Cop(M) (t) is cyclic by A.1.31.1. Thus if Mi is not
normal in M, then as the product MJ of the Mj is direct, MiM = M'f is of order
2, and mp(Mi) = 1, so that Mi falls into case (1) or (2) of D.2.17. In particular, if
mp(Mi) > 1, then Mi ::::] M.
Let Ki, ... ,Ka be the groups (MiM), and set wi := [V,Ki]; then MJ =Ki x
· · · x Ka and Vo = Wi EB··· EB Wa. Further V = Vo EB Cv(F(MJ)) by Coprime
Action.Assume first that J(T) 1:. CM(V). Then as M '/:-Mc, we conclude from 14.1.7
that either (1) or (3) holds, with (p, m) = (3, 1).
Thus in the remainder of the proof, we may assume that J(T) ~ CM(V).
Therefore since Mis maximal in M(T) under :S, we may apply 14.1.4 to conclude