16.4. INTERSECTIONS OF Na(L) WITH CONJUGATES OF Ca(L) 1185We claim that z is weakly closed in Z(T) with respect to G; the proof will
require several paragraphs. Suppose the claim fails. Then using Burnside's Fu-sion Lemma A.1.35, NG(T) induces Z 3 on Z(T), and in particular is transitive on
Z(T)#. Thus there are h, k E NG(T) such that v = zh and vz = zk; and in partic-
ular, NG(T) transitively permutes {To, T/!J, TM=: T. As K is tightly embeddedin G, distinct members of T intersect trivially.
Since TL and T/!: are normal in T, TL n T/!: is normal in T. Then as Z(T) n
TL = (v) is of order 2, v lies in TL n T/!: if this group is nontrivial; but this is
impossible as v = zh E T/!J and T/!J n T/!: = 1 by the previous paragraph. Therefore
[TL, Tf!:] ::; TL n T/!: = 1, so that T/!: ~ T/!:* ::; Gr• (TL). Now by 16.1.6, eitherGr.(TL) = (v) or H ~ Aut(A6) and Gr.(TL) = (v,x), where x induces a
transposition on Land L = (GL(v),GL(x)). Thus by 16.4.2.2, T/!:* =/= (v*,x*) when
H* ~ Aut(A 6 ), so [T/!:[ = 2. Therefore To= (z) is of order 2. Then zh = v E [T, T]
since L(r) ~ PGL2(q), so ash E NG(T), z E [T, T]. Thus [T: TL[ > 4, so that q = 9
and H* ~ Aut(A6)· Then [T*,T*] = Y*, where Z4 ~ Y::; TL, so [T,T]::; ToYand hence (v) = <l?([T, T]) :'SJ NG(T). This contradiction completes the proof of
the claim that z is weakly closed in Z(T) with respect to G. In particular, z rf_ vG.
By 16.4.2.4 there exists g E G with K' = K9 and Nr(K') ::; T9. By 16.4.3.2,
L'= [L',z].
We next establish symmetry between L, rand L', z by showing that L'(z) ~PGL 2 (q). Assume otherwise and recall E = (r, z) and TE E 8yh(GG(E)) is
abelian. Thus if q = 9, z does not induce an automorphism of L' contained in
86 • Therefore we may assume that z induces an inner automorphism on L'. By
16.4.2.1, R is Sylow in GK1(z), so as z E K'L' and R is of order 2, R is Sylow in
K'. Hence zL' nZ(T9) =/= 0, impossible as r is weakly closed in Z(T9) by the claim.
Therefore L'(z) ~ PGL2(q).
Recall that rv E rL, so if v E L', then by symmetry zv E zL', contrary to
the claim. Hence v rf_ L'. Recall that (z,r,v) = V = D1(TE) ~Es, and there is
t E NTL(TE) with [t,r] = v. As v rf_ L',
V n L' =: (u) =/= (v),
and by symmetry there is s E Nrg (TE) with [s, z] = u. Set X := NG(V) and
x+ := X/Gx(V), so that x+::; GL(V) ~ L 3 (2), and t+ ands+ are transvections
in x+ on V, with centers v, u, and axes Z(T) = (z, v), Z(T^9 ) = (r, u), respectively.
As Z(T) =/= Z(T9) and v =/= u, (t+, s+) is either D 8 or 83 from the structure of
L 3 (2). In the first case the unique hyperplane W of V normalized by (t+, s+) ~ Ds
is centralized by either t or s, say t; but then W = Z(T) is not centralized by
s, so that z =/= z^8 E Z(T), contrary to the claim. Hence (t+, s+) ~ 83 , and so
V = V1 EB Vi, where
V1 := (u, v) = [V, (s, t)] ~ E4,
and
Vi := Gv( (s, t)) = Z(T) n Z(T^9 ) = (vz) = (ur).
In particular as u is fused to ; and all involutions in D := TL(u) are in vG. As
Z rf_ VG by the claim, ZG n D = 0.
Next if To > (z), then Nr 0 (V)+ is a transvection on V with axis Z(T) and