671{3) Wo(R, V) S CT(V), and if n > 2 then W1(R, V):::; CT(V).
(4) Vu:= (VNa(U)/ is elementary abelian, and Vu /U E R 2 (Na(U)/U); further
[02(Na(U)), Vu] SU.
(5) Assume further that n > 2, and V^9 :::::; Ca(U) is V-invariant with [V, V9] -f- Then Ca(Zs) i M.
PROOF. Observe that R := CT(U) = Q(t), where Q := CT(V), and U and V/U
are the natural module for NL(U)/0 2 (NL(U)) s::! L 2 (2n/^2 ). Now A(R) = A(Q),
so J(R) = J(Q) ::::1 LT, and hence Na(R) :::::; Na(J(R)) :::::; M = !M(LT), so
RE 8yl2(Ca(U)). Similarly Tu := NT(U) E 8y[z(Na(U)) since J(Tu) = J(Q).
Thus (1) holds. As Un Z -f 1, F*(Na(U)) = 02 (Na(U)) by 1.1.4.3. Then
Ca(U) E 'He by 1.1.3.1, so (2) holds.
Next by 6.1.12, Wi := Wi(R, V) S CR(Zs) SQ for i = 0 when n ~ 2, and fori = 1 when n > 2. Thus (3) holds.
Let Vu := (VNa(U)) and Na(U)* := Na(U)/Ca(Vu). We may apply G.2.2
with U, V, 02 (CL(U)), Tu, Na(U) in the roles of "V1, V, L, T, H". ByG.2.2.4,Vu/U E R2(Na(U)/U). By G.2.2.1, Vu S 02(Ca(U)) and [02(Na(U)), Vu]:::::; U.
Then Vu:::::; 02(Ca(U)) SR using (1), so that Vu:::::; Wo(R, V) = Wo. Therefore as
W 0 :::; CT(V) by (3), Vu= (VNa(U)) is elementary abelian. This establishes (4).Now assume the hypotheses of (5). First m(V/Cv(V^9 )) = n, by applying
6.1.10.3 with the roles of V, V9 reversed. Then as U :::::; Cv(V9) with m(U) =
m(V/U) = n, we conclude U = Cv(V9). As n > 2, Lu := 02 (NL(U)) E.C(Na(U), Tu). As Tu E 8yl2(Na(U)) by (1), Lu S K E C(Na(U)) by 1.2.4.
As [U, Lu] = U, CK(U) S 000 (K). By (1) and a Frattini Argument, KR =
CKR(U)NKR(J(R)) = CK(U)(KnM)R. Now Lu= L[J ::::] KnM, and K/0 00 (K)is simple by A.3.3.1, so K = LuCK(U). Thus if CK(U) S M, then K S M, so
that K = K^00 =Lu. On the other hand, if CK(U) i M, then also 000 (K) i M.
By (3) and E.3.16, Na(Wo) S M ~ Ca(C1(R, V)). Each solvable subgroup
X of Ca(U) containing R satisfies n(X) = 1 by E.1.13, and so is contained in
M by E.3.19. This eliminates the exceptional case 000 (K) i M of the previous
paragraph, so that Lu= K. Since Tu normalizes Lu E C(Na(U)), and is Sylow in
Na(U) by (1), Na(U) normalizes Lu by 1.2.1.3. Then as 02(Lu) :::::; Q:::::; Ca(V),
02(Lu) S Ca(Vu).
Recall Vu is elementary abelian by (4). As Vis the direct sum of two copies
of the natural module U for Lu/02(Lu), and Lu ::::] Na(U), Vu is the sum and
hence the direct sum of copies of the natural module for Lu/ 02 (Lu). Next as
V9 s Ca(U), V9 S Rh for some h E Ca(U), so by (3)
V^9 s W 0 (Rh, V):::::; Qh:::::; 02(TDLu).
Thus [V^9 ,Lu]:::::; [0 2 (TJLu),Lu]:::::; 02(Lu) S Ca(Vu). Thus Lu normalizes Z1 :=
[V9Ca(Vu ), VJ = [V9, V]. We saw earlier that U = Cv(V9) with m(V/U) = n.
Then as n > 2, V:::::; 89, so that in fact 89 = V0 2 (£989). Hence Z1 = [V, V9] =
[8^9 , V^9 ] = Z~.
We finally assume that Ca(Zs) SM. Then Na(Zs) SM by 6.1.9.5, so