674 6. REDUCING L 2 (2n) TO n =^2 AND V ORTHOGONAL.
Then applying the subsequent arguments with Fin the role of "B", [F, Vo] =Zs
and Vo i. Q-y, so replacing 'Yo by o, we may assume that o ='Yo and Vo= V.
Let VB := V n B = Nv(E). Notice VB is of index at most 2 in V, as B
is of index 2 in UH. Then [VB,F] :::; Zs n Z-y, with VB, F of index 2 in V, E.
Therefore [VB, F] is of index at most 2 in Zs and Z-y by(!) and 6.1.17.4. Further
if [VB, F] = Zs, then Zs = [VB, F] = Z-y, which we saw earlier is not the case.
Hence [VB, F] is of index 2 in both Zs and Z-y, so IV: VBI = 2 by 6.1.17. Therefore
by 6.1.17.5, n = 2, and (z) := [VB, F] = Zs n Z-y is of order 2. Thus we have
established the first assertion of 6.1.19.
As DL is transitive on zf;, z is 2-central in LT, so we may assume T:::; Gz.
Thus H:::; Gz. As DL is transitive on zff, and Lis transitive on V#, we conclude
from A.1.7.1 that Gz := Cc(z) is transitive on the G-conjugates of Zs and V
containing z. Then Z-y = Z~ for g E Gz. Similarly if V:::; 02( Gz), then E:::; 02( Gz)
as EE v^0 •; but then E:::; 02 (Gz):::; 02(H), contrary to an earlier reduction. We
conclude Vi. 02(Gz).
Let W 0 := (V^1 ); to complete the proof, we assume W 0 is abelian and it remains
to derive a contradiction. Let Qz := (Z~z). By 1.1.4.6, F*(Gz) = 02(Gz). As
n = 2, [Zs,T]:::; (z), so Qz:::; 02(Gz) by B.2.14 applied in Gz := Gz/(z), and
hence Qz :::; T. Let W :=Won Qz; as I:::; Gz, I:::; Na(W). Set I* := I/C1(W).
Now Qz :::; T:::; Nc(V), so Qz :::; kera. (Na. (V)). Then as E E V^0 z, Qz acts on
E, and in particular W acts on E. We have seen that E :::; I:::; Nc(W), so that
[W, E] :::; W n E. Next as VB :::; Wo, [VB, E] :::; Wo. But Z-y :::; Qz as Z-y E z~·,
and Z-y = [VB, E] by (!) and 6.1.17.4, so Z-y :::; W n E. Finally if Z-y < En W,
then m(E/(E n W)) :::; 1 since n = 2. Then as V :::; W 0 and Wo is abelian by
assumption, V :::; Cc(E n W) :::; Cc(E) by 6.1.10.2, contrary to [VB, F] = (z).
Thus [E, W] :::; En W = Z-y, so [E, W] :::; Z-y of order 2, and hence E is trivial or
induces a group of transvections on w with center z'Y = z~.
Note that C1(W):::; Na(Z~):::; Na(02(F)), so that
Then as E :::; U-y :::; 02(1^9 ), but we saw E i. 02 (H), we conclude from (*) that
E does not centralize W, so that E* #-1. As W 0 is abelian, Z-y :::; C 1 (W), so we
conclude 1:::; m(E*):::; m(E/Z-y) = n = 2.
Let P := (E^1 ). As E centralizes Zs but NE(V) = F < E, Pi. M by 6.1.7.1.
As E:::; 02(1^9 ) and we saw C1(W) acts on 02 (J9), it follows from(*) that
[E, C1(W)]:::; 02(1^9 ) n C 1 (W):::; 02 (!),
so we conclude that Cp(W) :::; 02,z(P). Let Po denote the preimage in P of
02(P*). Then Po :::; 02,z,2(P) = 02,z(P), so that Po = 02 (P)Cp(W), and hence
02(P) = 02(P). On the other hand, by 6.1.17.2, 02 (P) :::; 02 (!) :::; C1(W 0 ) :::;
C1(W), so 02(P) = 02(P) = 1, and then W E R2(P). Thus as E induces a
group of transvections on W with center Z-y of order 2, we see from G.6.4 that P* is
the direct product of subgroups Xt isomorphic to Sm or Lk(2) for suitable m and
k. So either Xt ~ L2(2) ~ 83, or Xt is nonsolvable, in which case as the preimage
Xi is normal in P and Pis subnormal in Nc(Zs), Xi E C(Nc(Zs)). In that case,
as DL ~ Z3 and DL nI = 1, we conclude from A.3.18 that m 3 (Xi) = 1. Therefore