6.2. IDENTIFYING M22 VIA L2(4) ON THE NATURAL MODULE 685As Va i Nc(V(3), Va i G~~), so d(a) = b with a,"(,··· , 'Yl a geodesic, and we
have symmetry between"! and 'Yl· By this symmetry (as in the proof of 6.1.19) wecan apply F.7.13 to Va in the role of "A", to conclude that there exists fJ' E r('Y 1 )
such that m(Va/ Nv"' (Vf3')) = 1, and also that there exists h EH such that Vi fixes
fJ' and I:= (Ve,, Vi) is not a 2-group.
Observe next that ifμ, v E I'o, and Vμ acts on Vv, then [Vμ, Vv] = 1 by 6.2.7.
But as v; fixes (J', and fJ' = "16 for some g E G, Vi :::; Gf_ ::; Nc(Vf3'), so Vi
centralizes Vf3'. Similarly as Va does not centralize Vf3', Vf3' does not act on Va.
Thus fJ' satisfies the two conditions for "fJ" in our earlier argument, so we may
take fJ' =:= fJ. Then m(Va/Nv"'(V(3)) = 1, so that we have symmetry between a and
fJ. Thus as we showed that [Nv.e (Va), Va] = Cv"' (V(3) = Ua, by symmetry between
a and (J, [Nv"' (Vf3), Vf3] = Cv.e (Va)· In particular as Uf3 = Cv.e (Va), we also have
symmetry between Ua and Uf3. Further Nv.e (Va) and Nv"' (Vf3) are each of rank 3,
and induce a field automorphism on Va and Vf3, respectively. Hence
1 ::)= Ua,f3 := [Nv.e (Va), Nv"' (Vf3)] ::; Ua n Uf3.
Now Ua,f3 :::; Uf3 centralizes I as Va centralizes Ua and Vi centralizes Vf3' = Vf3.
Thus for Zo E ut,(3' Zo E Va' but Va i 02 ( G Zo )-since I ::; G zo ' and Va i 02 (I)
as I= (Va, Vi) is not a 2-group. As the pair (V, z) is conjugate to (V°" zo), 6.2.8
is established. D
In the remainder of this section, choose
H:=Gz,
and let Mz := CM(z), U := (Zf), K := (VH), MK := Kn M, and H* :=
H/CH(U). By 6.2.8, Vi 02(K), so Ki Nc(V). By 6.2.1, Nc(Zs) ::; Nc(V) =
Mv, and as Vis the natural module for L, CMv(z) ::; NM(Zs). As H = Gz, by
6.1.8 we conclude:
LEMMA 6.2.9. H n M = NH(V) = NH(Zs) and MK= NK(V) = NK(Zs).
LEMMA 6.2.10. (1) F*(H) = 02(H) =: QH and U::; Z(QH)·
(2) CH(U):::; Nc(V)::; M, so Cv(U)::; QH.
{3) 02(H*) = 1.
(4) V* # 1.
(5) [V,U]:::; vn u.
PROOF. The first assertion in (1) holds by 1.1.4.6. Hypothesis G.2.1 is sat-isfied with Z, Zs in the roles of "Vi", "V", so G.2.2 completes the proof of (1)
and establishes (3). By 6.2.9, CH(U) ::; NH(Zs) = NH(V) ::; M, so Cv(U) :::;
02 (CH(U))::; QH, proving (2). By (1), U is abelian, so by (2), U acts on V. Also
V::; H::; Nc(U), so (5) holds. As Vi QH by 6.2.8, (4) follows from (2). D
LEMMA 6.2.11. V* is of order 2.
PROOF. Assume the lemma fails; then as V ::)= 1 by 6.2.10.4, m(V) 2: 2. By
6.2.10.1, Zs ::; Cv(U), so that m(V) ::; m(V/Zs) = 2. Thus m(V) = 2, and
Zs = V n QH = V n U = Cv(U). Next by (4) and (5) of 6.2.10, 1 # [V*, U] ::;
vnu = Zs of order 2. Thus V induces a 4-group of transvections on [r with
center Zs. Also 02 (H) = 1 by 6.2.10.3. Thus we may apply G.3.1 and the results
of section G.6 to H*. In particular, since U = (Zf), we conclude from G.3.1 that