1146 15. THE CASE .Cf(G, T) = 0center Zs. Then V_H = (VH) is generated by transvections, tJ = (Zff), and
V :SJ T, so by G.6.4.4, VJ} = H ~ Ln(2), 2 :::; n:::; 5, 85, or 81, and U /Cu-(H)
is the natural module for H. As CH(Zs) is a 3'-group by 15.3.49.5, we conclude
that H* ~ 83. Then m(U) = 3 and Zs= Cu(V) = UnV. Now as 02(Y) = Cy(V)
by 15.3.49.3, [0 2 (Y), U] :::; Cu(V) :::; V; then in view of 15.3.51.3, Y centralizes
02 (Y)/V, so that V = 02 (Y). Thus Y ~ A4 x A4, contrary to 15.3.9. D
LEMMA 15.3.53. (1) UY= nt(V) = N1 x N2 with Ni~ 83 and v = [V,Ni]·
(2) V n QH = V n U = [U, VJ = ZJ_ is the hyperplane of V orthogonal to Z.
Thus V* is of order 2.
PROOF. By 15.3.52, conclusion (a) of 15.3.51.2 holds, giving (1). Next by (1),
[U, VJ = z1-, so as V* =f. 1 by 15.3.51.1, and [U, VJ :::; U n V by 15.3.51.4, (2)
follows. D
For the remainder of this subsection, define Ni as in 15.3.53, and set Yi :=
02 (Y n Ni),
LEMMA 15.3.54. Let g E Y with zg not orthogonal to Z in V, and set I :=
(U, UB), P := 02 (1), and W :=Un P. Then
(1) I= YU.
(2) P = WWB and V:::; Z(P).(3) u n UB = w n WB = z1-n zgj_ ~ E 4.
(4) P/V = PifV EB P2/V, where PifV := [P/V, Yi]= Cp;v(N3-i), and Pi/V
is the sum of s natural modules for Ni.
(5) [WB, U]:::; W and WB normalizes U.
PROOF. We verify the hypotheses of G.2.6, with V, Y, Z, U in the roles of "VL,
L, V 1 , U" By 15.3.481, G.2.2 is satisfied by the tuple of groups, and the remaining
hypotheses of G.2.6 hold by 15.3.53. Hence the conclusions of G.2.6 hold with
V(U n UB) in the role of "8 2 ". Thus conclusions (1) and (2) of 15.3.54 follow from
G.2.6, and conclusion ( 4) will follow from G.2.6.5 once we show that Un U^9 :::; V.
As WB :::; T, WB normalizes U, so [WB, U] :::; P n U = W, and hence (5) holds.
Further' <J?(UB) :::; zg by 15.3.48.2, so [Un UB, WB] :::; w n Z9. But zg :::; v, so
wnzg:::; unv, and hence wnzg = 1, since we chose Z^9 i z1-, and z1-= unv
by 15.3.53.2. Thus WB centralizes u n UB' and by symmetry, w centralizes u n UB'
so using (2), Po := (Un UB)V:::; Z(P). Further by G.2.6.4, I centralizes Po/V, so
since Po:::; Z(P), we may apply CoprimeAction to conclude Po= VxCp 0 (Y). Now
T normalizes YU= I, and hence normalizes the preimage Po of Co 2 (I)/v(Y) in I,
and then also normalizes Cp 0 (Y). Therefore as 01 (Z(T)) = Z:::; V, we conclude
Cp 0 (Y) = 1 so that Po = V, and hence Un UB :::; V. As mentioned earlier, this
completes the proof of (4), and we established (5) earlier, so it remains to complete
the proof of (3). But by 15.3.53.2, Un V = ZJ_, so as Un UB :::; V,
u n U^9 = (Un V) n (U^9 n V) = zj_ n z^9 _1_ = w n W^9 ~ E 4.
DIn the next few lemmas, we use techniques similar to those in section 12.8 to
study the action of H on U.
For the remainder of the subsection, define g, W, P, Pi, ands as in 15.3.54.
LEMMA 15.3.55. U is extraspecial, and V = Z(P).