12.7. THE TREATMENT OF As ON A 6-DIMENSIONAL MODULE 841x EX, Kt"' = K'f:::; Ca(vt"') = Ca(Vt):::; Gt, so that Kt= Kt"'· Hence X acts on
Kt and XKt/Z(Kt) ~ PGL3(4).
From the structure of Kt, Lt is an L 2 (4)-block, so L is an A 6 -block. Let
Yx E Syl3(X). As X = 02 (02,z(L)) and L is an A 6 -block, X = VYx. As
Kt :::l Gt and S11(02(Kt)) =Vt, Gt :::; Na(Vt). Then as Xis transitive on v;,#, Vt
is a TI-set in G by I.6.1.1.
Next as V:::; Lt by 12.7.4.2, Ca(Lt):::; Cat(Lt) = Cat(Kt), so as CAut(Kt)(Lt) =
1, Ca(Lt) = Cat (Kt). Similarly Ca(Lt) :::; Ca(V) = CM(V), and [L, CM(X)] :::;CL(X) = Z(L), so as L is perfect, Ca(LtX) = Ca(L) is LT-invariant. Further
[Ca(Lt), X]:::; Cx(Lt) =Vt, so by a Frattini Argument, Ca(Lt) = VtCa(LtYx) =
Ca(LtX). On the other hand, we saw that Ca(Lt) =Cat (Kt), so if Ca(LtX) -I-1,
then Kt :::; Ca(Ca(Lt)) :::; M = 1M(LT), contrary to Kt i Mt. ThereforeCa(LtX) = Ca(L) = 1, and Cat (Kt) = Ca(Lt) = VtCa(LtX) = Vt. Thus
V = 02(M) and M = LT by 12.7.6.2. Then by 12.7.2.1, either M = L, or
[M: L[ = 2 with M/V ~ S5.
Choose t so that Tt := Cr(t) E Syl2(Mt)· As Kt :::l Gt, Gt~ 1-le, sot is not2-central in G by 1.1.4.6. hence P = Cy(F) since Inn(P) induces CAut(P)(P) by
A.1.23. ThereforeY/P:::; Aut(P) ~ ot(2),
and Ds ~ T/P E Syh(Y/P). Further a:= (Mz/P,T/P,Naz(U)/P) is a Gold-
schmidt triple as in Definition Aa.t:defnGldtrpl. As 02 (Mz/P) -I-02(Naz(U)/P),
case (i) of F.6.11.2 holds, and so the image in Y/03'(Y) of a is a Goldschmidt
amalgam; therefore as Y is an SQTK-group, Y/0 3 1(Y) is described in Theorem
F.6.18. In view of(*), Y/031(Y) appears in case (6) of Theorem F.6.18; that is,
Y/0 3 ,(Y) ~ L 2 (q) for q = ±7 mod 16. Then as Y/P :::; Ot(2), we conclude
Y/P9!:L3(2) orA5..Next P+ is the sum of the natural module and its dual for y+ / p+ 9!: L 3 (2), so
M; and Na;:(U+) stabilize uniqμe points of P+. Indeed i\+ is the point stabilized
by M;, and we write fJt for the point stabilized by Na;:(U+). Applying ((J, Mz
stabilizes only Vi and Na.(U) stabilizes only U1. As if 1 + -I-tJt, V1 -I-U1. But if
Y/P ~ A 6 , then Y stabilizes a point of P, so V1 = Cp(Y) = U1, contrary to the
previous remark. We conclude Y/P 9!: L3(2).
Now 84 9!: Mz/ P is the stabilizer in Y/ P of V 1 , so P 1 := (°t\Y) is a nontrivialquotient of the 7-dimensional permutation module on Y/Mz, and similarly P 2 :=
(U[) is a nontrivial quotient of the permutation module on Y/Naz (U). Hence by
H.5.3, either P =Pi is the 6-dimensional core of the permutation module for i = 1
or 2, or else P = Pi EB P 2 with dim(Fi) = 3 for i = 1 and 2. Next '-P : T+ -+ T
is an isomorphism, and for each 3-dimensional indecomposable W for a rank one
parabolic y 0 + of Y+ containing the fixed point of Y 0 +, P+ splits over W as a T+ -
module. However this is not the case when Pis the core of the permutation module,and that module is indecomposable. Hence the former case is impossible, so the
latter holds.
Now Qz:::; Pis Y-invariant, so Qz = (z), Pi, or P. As F*(Gz) =. Qz, the first
· case is out. Next suppose Qz =Pi. Now Pi 9!: El6, and as TE Syh(G) normalizes
Pi, Na(Pi) E 1-le by 1.1.4.6, so Ca(Pi) E 1-le by 1.1.3.1. Therefore as Cr(Pi) =Pi,
we conclude Ca(Pi) = Pi. Now GL(Pi) = Aut(Pi) with Y/Pi = CaL(Pt)(z),
so Gz = YCa(Pi) = Y normalizes P, contrary to 02(Gz) = Qz =Pi < P. Thus