838 i2. LARGER GROUPS OVER F2 IN Cj (G, T)Suppose now that [Ui, K2] = 1. Then Vs = V5,i EEl Vs,2, where V5,i := [il5, Xi] ~
E 4. Then, as in the proof of 12.6.32, 115,i contains a nonsingular point ui, andK 3 _i ::; Cc(ui) ::; M by Theorem 12.6.2, contrary to Ko </: M.
This contradiction shows [Ui, K 2 ] =I 1. Suppose now that case (a) holds. Then
[Ui,K2] = [UH,Ki,K2]::; [IH,Ki]::; CuH(K2). So [Ui,K2] = [Ui,K2,K2] = 1,
contrary to [Ui, K2] =I 1.
Therefore case (b) holds. As in the proof of 12.6.32, Ri = T* n K 0 and V5 ::;
CoH(Rt). But as Uk is the nt(4)-module, CoH(Ri) is an F4-point of UH, whereas
E15 ~ V5 ::; CoH (Ri). This contradiction completes the proof of 12.6.33. DWe are at last in a position to obtain a contradiction under the hypotheses of
this section.
By 12.6.33, H* = Hi x H2 with Hi ~ 83 and H2 ~ 8s. In particular
LiT/02(LiT) ~ 83 x 83, so f' ::; L and Mv = L ~ As. Also X E 8y[s(Li)
is of the form X =Xi x X2 with Xi E 8y[s(Hi)· As XiT = TXi, we conclude
each Xi moves 6 points of O; hence Cv(Xi) is a nondegenerate space of dimension
- -H - -
2 and Vs= [Vs, Xi]· In particular as Xi ::;! H* and UH= Ws ), UH= [UH,Xi],
and then
- -H - -
UH= Ui EEl U2
is an H2-decomposition of u H' where ui := [UH' ti] for ti and t:j distinct involutions
in Hi. As the third involution t3 in Hi commutes with H2 while interchanging Ui
and U2, and F(H2) = K is simple, H2 is faithful on Ui, and U 2 is isomorphic to
U1 as an H2-module. Recalling that H2 ~ 8s has no strong FF-modules, we must
be in case (b) of F.9.18.6, so that Ut is an FF-module for H2. Hence either [Ui, H2]
is the 8s-module, or [Ui,H2] is the L2(4)-module, where UH:= UH/CoH(K).
In particular, no member of H* induces a transvection on UH. Thus by F. 9.16.1,
D'Y < U'Y, in the notation of F.9.16. Hence by F.9.16.4, we can choose"! so that
0 < m := m(U;) ~ m(UH/DH)· Further by F.9.13.2, U'Y :S 02(G'Yl.'Y 2 ) =Rf for
suitable h EH, so m :S 2 as H* ~ 8s x 83. Next for b E U'Y - D'Y, [DH, b] :S Ai by
F.9.13.6, where Ai is the conjugate of Vi defined in section F.9; thus m(Ai) = 1
and
m([UH, b*]) ::; m(UH /DH)+ m([DH, b]) ::; m + 1::; 3,
impossible as m([UH,b]) = m([Ui,b]) +m([U2,b]) = 4, since b Eu;::; Rf::; K*
and Ui is the natural or As-module for K*.
This contradiction finally eliminates the A 8 -subcase of Theorem 12.2.2.3d, and
hence establishes:
THEOREM 12.6.34. If Hypothesis 12.2.3 holds with L ~ L 4 (2) 1 then V is the
natural module for L.
12.7. The treatment of A 6 on a 6-dimensional module
In this section we prove
THEOREM 12.7.1. Assume Hypothesis 12.2.3 with L/CL(V) ~ A 6. Then G is
isomorphic to M24 or He.