1086 15. THE CASE .Cf(G, T) = r/JThus we may assume that case (i) holds, so MJ = K1 and V =Vo= W1. Thus
Mis faithful on W1 and M ~ NaL(W 1 )(K1).Assume first that (MJ, V) is indecomposable. Then s = 1, so MJ = M1 and
V = V 1. Recall we showed that conclusion (6) of D.2.17 does not hold for (MJ, V).
Conclusions (1) and (2) of D.2.17 give conclusion (1) of 15.1.2.
Suppose conclusion (5) of D.2.17 holds. Then MJ = n;t(2), so M = MJ, for
otherwise M = NoL(v)(MJ) = O;t(V) contains transvections, whereas ij(M, V) =3/2 in case (5) of D.2.17. Thus conclusion (2) of 15.1.2 holds in this case.
Suppose conclusion (3) of D.2.17 holds. We showed earlier that M ~ O;t(V)
and M acts irreducibly on V. As conclusion (3) of D.2.17 holds, ij(M, V) = 2, so
f' contains no transvections on V. Hence as M is irreducible on V, f' ~ Z4, so
conclusion (5) of 15.1.2 holds.Suppose that case (4) ofD.2.17 holds. Then MJ = P([/, where P = F*(MJ) ~
3i+^2 , f inverts P/if!(P), and m(V) = 6. Hence M:::; NaL(V)(MJ) ~ GL2(3)/31+^2.
If 02 (M) > P, then m 3 (Cf.iJ(f)) > 1, contrary to A.1.31.1; thus 02 (M) = P.
Therefore ifT is irreducible on P/if!(P), then conclusion (4) of 15.1.2 holds, so we
may assume that Tis reducible on P /if!(P), and it remains to derive a contradiction.
Then f' ~ Z 2 or E 4 , and in either case Tacts on subgroups P 1 and P2 of order 3
generating P. Thus Z = EiE 2 , where 1 =f. Ei := Cv(PiT). Therefore the preimages
Pi satisfy PiT ~ Ca(Ei) ~Mc= !M(Ca(Z)), and hence M = (P1,P2)T ~Mc, a
contradiction.
Finally assume that (MJ, V) is decomposable. As (i) holds, a= 1. Then fromthe second paragraph of the proof, s = 2 and (Mi, Vi) satisfies case (1) or (2) of
D.2.17. As a= 1, M 1 and M 2 are interchanged in M, so that conclusion (3) of
15.1.2 holds.
This completes the proof of 15.1.2. D15.1.1. Statement of the main theorem, and some preliminaries. Our
first goal is to show that case (1) or (3) of 15.1.2 holds, and V(M) is an FF-module
for M. That is, we will prove that either m(V(M)) = 2 with M/CM(V(M)) =
GL(V(M)) ~ L2(2), or m(V(M)) = 4 with M/CM(V(M)) = ot(V).
In the remaining cases there are no quasithin examples; indeed as far as we
can tell, there are not even any shadows. But we saw in Theorem 14.6.25 ofthe previous chapter that quasithin examples do arise in the first case, and many
shadows complicate our analysis of the second case, in the third section 15.3 of this
chapter.Thus the remainder of this section is devoted to the first steps in a proof of the
following main result:THEOREM 15.1.3. Assume Hypothesis 14.1.5, and let M := MJ as in 14.1.12.
Then either(1) m(V(M)) = 2, M/CM(V(M)) ~ £2(2), and G is isomorphic to J 2 , J 3 ,
(^3) D 4 (2), the Tits group (^2) F 4 (2)', G 2 (2)' ~ U3 (3), or Mi2; or
(2) m(V(M)) = 4, and M/CM(V(M)) = O;t(V(M)).
The proof of Theorem 15.1.3 involves a series of reductions, which will not be
completed until the end of section 15.2. Thus in the remainder of this section, and
throughout section 15.2, we assume G is a counterexample to Theorem 15.1.3. We
also adopt the following convention: