914 13. MID-SIZE GROUPS OVER F2
section including Definition F.7.2, where in particular "(o, "(1 are the cosets G1, G2.
For 'Y = "fog set V-y := V9, while for 'Y = "f1g set V-y := Vcf. Let
be a geodesic in r, of minimal length n subject to Va i. G~
1
); such an n exists
by F.7.3.8. As G~^1 ) = 02 (Gf3) = Ca!'l(Vf3) for each /3 E r, [Va, Vf3] # 1, and so
we have symmetry between a and /3. This symmetry is fairly unusual among our
applications of section F. 7, as we almost always consider only geodesics whose orgin
is conjugate to 'Yoi however the approach in this lemma is the one most commonly
used in the amalgam method in the literature. By minimality of n,
Va :<:::; G~^1 2_ 1 :<:::; Gf3,so Va acts on Vf3, and by symmetry, Vf3 acts on Va. By F.7.9.1, Va :<:::; 02 ( Ga,an-i (1) ) ,
and
02(G~~n-J = 02(Ho n M) = 02(L2T)^9 = R~,
for g E G with {'Yog,"(1g} = {an-1,/3}
using transitivity of (G 1 , G 2 ) on the edges of the graph in F.7.3.2. Thus Va :<:::; R~.
Suppose first that /3 = "( 1 g; then V/3 = Vcf. If a is conjugate to 'Y1 then wemay take a = 'Yo and a 1 = "fl, so V 0 :<:::; R~, and by (*) and symmetry between
a and f3, Vcf = V/3 :<:::; R 2 • As 1 "I [Va, Vf3] = [Vo, Vcf], we have a contradiction to
13.4.16. Thus at most one of a and /3 is conjugate to 'Yl· Therefore if /3 tf-"(oG,
then a E 'YoG, so reversing the roles of a and /3, we may assume /3 E "(oG. Thenconjugating in G, we may take /3 := 'Yo and C¥n-1 := "fl· Thus Va :<:::; R2 by (*).
Similarly a= "fig for i = 0 or 1, and by (*)we may take V :<:::; R~.
If a = "( 1 g then Va = Vcf, contrary to 13.4.15. Hence a = "(og and Va = V^9.In particular 1 -=f. [V, V9] :<:::; V n V9.
Let v E [V, V9]#. If Ca(v) :<:::; M, then by 13.4.2.3, Vis the unique member of
v^0 containing v; hence Ca(v) i. M. However for any t E T-CT(V), [V, t] contains
a vector of weight 2, so z is of weight 2 by the uniqueness of z in 13.4.12.1; thus (1)
holds. Indeed by (1) and that uniqueness, Ca(w) :<:::; M for w EV of weight 4, and
hence V is the unique member of va containing w by 13.4.2.3. This establishes(6).
By (6), all vectors in [V, V9]# are of weight 2, so [V, V9] is of rank 1-since up
to conjugacy, E := (e5, 6 , e4, 6 ) is the unique maximal subspace of V with all nonzero
vectors of weight 2, and E -=f. [V, A] for any elementary 2-subgroup of M. Then
conjugating in £ 2 :<:::; Gf3, we may assume [V, V9} = (z). Now by 13.4.2.3, we may
take g EH, so (2) is established.
As z is of weight 2 in V, we are in case (ii) of 13.4.14.2. Hence either (3) holds,
or else VH is a 5-dimensional module for H/0 2 (H) ~ A 6 and Zv of weight 4 in VH.But in the latter case we have symmetry between L, V and K := 02 (H), VH, so
as Zv is weight 4 in VH, we have a contradiction to (1) applied to K, VH·. Hence
(3) is established. By (3), VH is not a 5-dimensional module for K/Cx(VH) ~ A5,
and in the remaining two cases in 13.4.13, VH is the natural module for K/02(K),
so His transitive on the points of VH; thus (4) is established as Zv :<:::; VH by (3).
If U :<:::; V with Ca(U) i. M, then all vectors in U# are of weight 2 by (6). But