soo 12. LARGER GROUPS OVER F2 IN .Cj(G, T)
(1) Mv = Gv ~Gu= Mu.
Next if u"' = v for some x EM, then gx E Gv:::::; M, so g E Mx-^1 = M, contrary
to the choice of g. Hence
(2) Utt VM.
By (1) and (2) there are u, v E V# with Mu ~ Mv but v tt uM. Inspecting the
list of Theorem 12.2.2.3, we first eliminate the cases where L is irreducible on V.
In the remaining cases, let z denote the generator of Cv(L). We also eliminate
case (b), since there Zn V = Cv(L) by B.4.8.2, so that each v E V - Cv(L) is
T-conjugate to vz, and hence all members of V -Cv(L) are M-conjugate. This
leaves the subcases of ( d) where V is the core of the permutation module of degree
n for L ~An, n = 6 or 8, and the subcase of (f) where V is the Weyl module of
dimension 7 for L ~ G 2 (2)'. In the former, we may take v of weight 2, and u of
weight n-2; in the latter, we may take v singular and u nonsingular in V -Cv(L).
Now conjugating in L, we may assume u = vz. As Cv(L) =/= 1, V ~ M by 12.2.2.3,
so z E Z(M) and hence M = Gz.
Without loss, Tv := CT(v) E Syb(Mv), so as Gv :::::; M, Tv E Syb(Gv)· As
u = vz with z central in Mu = Gu, also Tv E Syb(Gu)· Then replacing g by a
suitable member of gCa(u), we may assume g E Na(Tv)· However if L ~ A6 or
G2(2)', then vis 2-central in M so that T = Tv, so g E Na(T) :::::; M by Theorem
3.3.1, contrary to (2). Hence L ~As.
Let Zv := fh(Z(Tv)) and Vv := (Z[:). As Q :::::; Tv, Q centralizes Zv and
hence Vv· On the other hand, Zv contains Zn V i. Cv(L) by I.2.3.li; so V :::::;
Vv as V E Irr +(L, R2(LT)), and hence CLT(Vv) :::::; CLT(V) = Q. Therefore
Q = CLT(Vv), and hence Vv E R2(LT). If [Vv, J(T)] = 1, then as V :::::; Vv, also
[V, J(T)] = 1, and then 3.2.10.2 contradicts (2). Thus [Vv, J(T)] =/= 1, so by B.2.7,
Vv is an FF-module for LT/CLT(Vv). Then as Vis the core of the permutation
module for As, by Theorem B.5.1.1, V = [Vv,LJ, and hence Vv = VZL by B.2.13,
where ZL := Cz.,,(L). Thus Zv = (v)ZL. Now T = Tv(T n £) :::::; Ca(ZL), so
ZL n Zf = 1 using (2), since M = !M(LT) and MB = !M(£9TB). Hence as
g E Na(Tv):::::; Na(Zv) and ZL is a hyperplane of Zv, we conclude ZL is of order 2.
Therefore ZL = (z) and Zv = (z, v). Then as vB = u rf. zG by (1), z is weakly closed
in Zv. Therefore g E Gz = M, contrary to (2), completing the proof of 12.2.16. D
LEMMA 12.2.17. W 0 (T, V):::::; CT(V) = Q, so that Na(W 0 (T, V)):::::; M.
PROOF. Let VB :::::; T. By 12.2.16, V is a TI-set in G, so as VB = Nvg (V),
we conclude from I.6.2.1 that [V, VB] = 1. Now the final statement follows from
E.3.34.2. D
During the remainder of the proof of Theorem 12.2.13, pick HE 1i*(T,M),
and set K := 02 (H), VH := (ZH) = (ZK), and H* := H/CH(VH)· Observe that
Ca(Z):::::; Ca(Z n V) :::::; Na(V):::::; M by 12.2.16; thus we may apply 12.2.7 during
the course of the proof.
LEMMA 12.2.18. Vt;_ 02(H), so that V* =/= 1.
PROOF. By 12.2.7.1, 02 (H) = CT(VH)· Thus V:::::; 02 (H) iff V* = 1, so we
may assume V:::::; 02(H), and it remains to derive a contradiction.
Similarly if Wo := Wo(T, V) :::::; 02(H), then H :::::; Na(Wo) :::::; M by E.3.15
and 12.2.17, contrary to H i. M. Thus there is A := VB :::::; T, with A* =/= 1, and