946 13. MID-SIZE GROUPS OVER F2so L 1 has at least six noncentral 2-chief factors. Therefore m(A*) 2 4 by 13.7.10.8.
On the other hand as EndK(UH/CoH(K)) ~ F2, we conclude K* = F*(H*); so A*
acts faithfully on K*, and hence m(A*) :S: m 2 (Aut(K*)) = 3. This contradiction
completes the proof of Theorem 13. 7.8.12.7. The treatment of A 6 on a 6-dimensional module
In this section, we complete the treatment of A5. We prove:THEOREM 13.8.1. Assume Hypothesis 13.3.1 with L/02,z(L) ~ A5. Then G
is isomorphic to Sp5(2) or U4(3).Throughout this section, we assume that G is a counterexample to Theorem
13.8.1.
Since L/0 2 ,z(L) ~ A 6 , we continue with the notation established in section
13.5: Namely we adopt the notational conventions of section B.3 and Notations
12.2.5 and 13.2.1.
As G is a counterexample to Theorem 13.8.1, G is not isomorphic to U4(3)or Sp 6 (2). Thus Hypotheses 13.5.1 and 13.7.1 hold, so we may apply results from
sections 13.5 and 13.7. In particular recall from 13.5.2.3 that Vis the 4-dimensional
A 6 -module. The main result Theorem 13.7.8 of section 13.7 has reduced us to the
following situation (where Hz is defined below):
LEMMA 13.8.2. (V^01 ) is abelian, so VH is abelian for each HE Hz.
As in the previous section, there are no quasithin examples under this restric-
tion, so we are continuing to work toward a contradiction. Again as far as we can
tell, there are not even any shadows.
LEMMA 13.8.3. If g E G with 1 # V n Vg, then [V, Vg] = 1.
PROOF. As Lis transitive on V#, G 1 is transitive on conjugates of V containing
Vi by A.1.7.1, so we may take g E G 1. Then (V, V9) :S: (V^01 ), so the result follows
from 13.8.2. 0
As usual z is a generator for Vi, and as in Notation 13.5.8, Ch := GifV 1 • By
13.3.6, G1 1. M, so Hz# 0, where
Hz:= {HE H(L1T): H :S: G1 and H 1. M}.For the remainder of the section, let H denote some member of Hz.
By 13.5.7, Hypothesis F.9.1 is satisfied with Vs in the role of "V+". From
Notation 13.5.8, UH:= (Vl/, VH := (VH), QH := 02(H) = CH(UH), and H* :=
H/QH so that 02(H*) = 1. Furthermore set He:= CH(UH); then He :S: QH.
Now condition (f) of Hypothesis F.9.8 is satisfied by 13.8.3, and condition
(g.i) of Hypothesis F.9.8 is satisfied since [V, CH(V 3 )] :S: Vi by 13.5.4.4; indeed
C.111v(Vs) :S: ((5,6)), with (5,6) inducing the transvection on V with center Vi.Thus we can appeal to the results in sections F.7 and F.9. In particular, we
form the coset geometry r of Definition F.7.2 on the pair of subgroups LT and H,
and let b := b(r, V). Choose 'YE r with d('Y 0 , "!)=band V 1. G~^1 ). By F.9.11.1, bis odd and b 2 3. Without loss ')'1 is on the geodesic
/'o, 'Yl, · · · , /'b := /'
from ')'o to ')'.