632 5. THE GENERIC CASE: £2(2") IN .CJ AND n(H) >^1
(2) n = 2 or 4, [V, L] is the natural module for L, and [Z, H] = 1.
(3) n = 2, [V, L] is the 85 -module for LT~ 85, and Z(H) = l.
(4) n = 0 mod 4, v is the rr4(2nl^2 )-module for L, and [Z,H] = l. Fur-
thermore if we take DE to be the subgroup of DL of order 2n/^2 - E, E = ±1, and
Xe:= (De, H), then Z::::; Z(JL) and either 02 (X+)-=/:-1, or n = 4 or 8.
PROOF. Let X := (DL,H). Then by hypothesis, 02(X) = l. Recall from the
start of the chapter that Z = Q 1 (Z(T)), and set VD := (ZDL) and Vz := (ZL).
Observe that Vz E R 2 (LT) and VD E R 2 (TDL) by B.2.14. In each case of 5.1.3,
v =((Zn V)L)::::; Vz.
Suppose first that T :'.SI TD£. Then applying Theorem 3.1.1 with TDL, Tin
the roles of "Mo, R", we contradict 02 (X) = 1. Therefore TS TDL·
Since L ~ L 2 (2n), it follows that n is even, and also that LT = LS where
8::::; T, S-=/:-1, L n S = 1, and S acts faithfully as field automorphisms of L.
As Vz E R 2 (LT), we can apply 5.1.2 and 5.1.3 to Vz in the role of "V". For
example by 5.1.2 and 3.1.8.3, either
(i) [Z,H] = 1 = Cvz(L), or
(ii) [Vz, J(T)] -=1-1, and either Vz/Cvz(L) is the natural module for L, or [Vz,L]
is the 85-module for LT~ 85.
To complete the proof, we consider each of the possibilities for V arising in 5.1.3.Suppose first that V is described in case (1) of 5.1.3. As the overgroup Vz of
Vis also described in one of the cases in 5.1.3, we conclude that V = Vz. By the
previous paragraph, [Z, H] = 1. From the structure of V, VD ::::; Cv(T n L) which
is of rank 2n in V of rank 4n, DL is faithful on VD so that m(VD) 2: n, with
(TnL)Cr(V) = 02(TDL) = Cr(VD) = CrDL(VD),
and T/Cr(VD) is cyclic. Thus as H n M normalizes TDL by 5.1.5.1, Hypothesis3.1.5 is satisfied by TDL, VD in the roles of "M 0 , V". As 02 (X) = 1, we conclude
from 3.1.6 that q(TDL/02(TDL), VD) ::::; 2. Hence as T/Cr(VD) is cyclic and
m(VD) 2: n, we conclude that n = 2, so that conclusion (1) holds.
Similarly if V appears in case (3) of 5.1.3, we conclude as in the previousparagraph that Vz appears in case (1) or (3) of 5.1.3, that Hypothesis 3.1.5 is
satisfied with TDL, VD in the roles of "Mo, V", and that q(TDL/02(TDL), VD)::::;
- Hence either n = 2, or possibly n = 4 in case Vz satisfies conclusion (3) of 5.1.3-
since m(VD/Cvn(t)) = n/2 fort E T-Cr(VD) with t^2 E Cr(VD) when Vz satisfies
that conclusion. Further J(T) ::::; Cr(VD) by B.4.2.1, so [H, Z] = 1 = Cz(L) by
Theorem 3.1.7, which completes the proof that conclusion (2) holds in this case.
Suppose next that V appears in case (2) or (5) of 5.1.3, or in case (4) with
n = 2. These are the cases where n = 2 and L has an A 5 -submodule on V,
and hence also on Vz, so that Vz must also satisfy one of these three conclusions.
Therefore DL ::::; Ca(Z). Recall H E ?-l(T) ~ ?-le by 1.1.4.6, so if Z(H) -=/:- 1 then
Zn Z(H)-=/:-1. Thus as 02(X) = 1, Z(H) = 1, so that case (ii) holds; therefore Vz
satisfies conclusion (3), and hence so does V.
This leaves the case where V satisfies case ( 4) of 5.1.3 with n > 2. Thus V = Vz
as before, and hence (ii) does not hold, leaving case (i) where [Z, HJ= 1 = Cz(L).
Now Vis a 4-dimensional FL-module, where F := F 2 ,,,;2, and Z = Cu(T) where U
is the 1-dimensional F-subspace of V stabilized by S := TnL. Further setting A:=