928 13. MID-SIZE GROUPS OVER Fz02(Lv)I = 3, we conclude IB : Be02(B)I = 3 and Be -=f 1. Then Lv/02(Lv) is
inverted in Tv n L :::; Crv (Be/ 02 (Be)). This is impossible since each element inAut(Sp 4 (2m)) acting on B and inverting an element of.order 3 in B* induces a
field automorphism on K* inverting fh(0 3 (B* /0 2 (B*))) ~ Eg. This contradiction
completes the proof of 13.6.6. DWe come to the main technical result of the section, requiring the bulk of the
argument; afterwards the remainder of the proof of Theorem 13.6.1 is fairly brief.THEOREM 13.6.7. (1) Gv:::; Mv. Hence Gv = CMv(v).
(2) Na(Vv) :::; M.
Until the proof of Theorem i3.6. 7 is complete, assume G is a counterexample.
We begin a series of reductions.Recall V,, = (v) x [V, Lv] with { v} U [V, Lv]# the set of nonsingular vectors in
Vv. Therefore by 13.2.6.2, Na(Vv) acts on the three singular vectors of Vv, and
hence preserves their product v-so that Na(Vv) :::; Gv, and hence (1) implies (2).
On the other hand, if V,, :sJ Gv, then Gv permutes
V := {Vu : u E Vv and u is nonsingular},
so Gv acts on V = (V), and hence Gv:::; Na(V) = Mv, so (1) holds. Thus we may
assume:LEMMA 13.6.8. Gv > Mv and Vv is not normal in Gv.
Set Uv := (z^0 v) and G~ := Gv/CaJUv)· Regard G~ as a subgroup of GL(Uv)
and write ux* for the image of u E Uv under x* E G~.As Lv :::; Gv, V,, :::; Uv. As z E Z(Tv), by 13.6.6 we may apply B.2.14 to
conclude Uv E R2(Gv)· In particular 02(G~) = 1 and Uv:::; Z(Qv)·
If [Uv, J(Tv)] = 1, then S = Baum(Crv(Uv)) by B.2.3.5 and 13.6.4, so by a
Frattini Argument and 13.6.5,Gv = Cav(Uv)Nav(S):::; Ca(Uv)CMv(v).
But then VvGv = V,;Mv = {Vv}, contrary to 13.6.8. Hence
LEMMA 13.6.9. J(Gv)* -=f 1.
Thus Uv is an FF-module for G~.LEMMA 13.6.10. If L~ = [L~, J(Tv)*], then L~ is not subnormal in. G~.
PROOF. Suppose otherwise. Then 02 (L~):::; 02 (G~) = 1, so that L~ has order
- Further m([Uv,L~]) = 2 by Theorem B.5.6, so [Uv,Lv] = [V,Lv], and hence
V,, = (v) X [V, Lv] = (v)[Uv, Lv]. Now Theorem B.5.6 also shows that IL~G~ I :::; 2, so
as Lv is Tv-invariant, L~ is normal in G~, so that (v)[Uv, Lv] = V,, is Gv-invariant,
contrary to 13.6.8. D
Let Xo be the set of L~T:-invariant subgroups X = 02 (X) of G~ such that
1 -=f X = [X, J(Tv)*]. Let X denote the set of all members of X 0 normal in
G~, and Xz the set of those X in X 0 with [z, X] -=f 1. For X* E X 0 , set U x :=
[(zX*L~),X*].