6.2. IDENTIFYING M22 VIA L2(4) ON THE NATURAL MODULE 679in [Asc86a]. Therefore to prove (2), it remains to show Fu := Cp(u) = 1-since
this shows IFI ::::; 4, and we saw earlier that E =VF.
As L =Lo, we may take DL ::::; Lo ::::; Ca(F), so DL ::::; CL(Fu)· Thus as DL
is irreducible on (Un R)/E and u E (Un R) - E, Un R centralizes Fu- ThenR ::::; (UL^0 ) ::::; Ca(Fu), so L ::::; LoR ::::; Ca(Fu), and hence Fu = 1 by 6.1.6.1, as
desired. DWe now complete this section by eliminating case (1) of 6.1.15-hence reducing
Hypothesis 6.1.1 to the case leading to M 22 in the following chapter:THEOREM 6.1.27. Assume Hypothesis 6.1.1 and set VL := [V,L]. Then
(1) n = 2.
(2) VL is the natural module for L/0 2 (L) ~ L 2 (4) and CT(L) = 1.
(3) Let Zs:= CvL(TL)· Then Ca(Zs)::::; M.
(4) Either Na(Wo(T, VL)) i Mor W1(T, Vi) i CT(Zs).PROOF. By 6.1.6.2, VL is the natural module for L/0 2 (L) ~ L 2 (2n), and
CT(L) = 1 by 6.1.6.1. Thus to complete the proof of (2), it suffices to prove (1).
As the statements in Theorem 6.1.27 concerning V are about VL, we may as
well assume V = VL, so that we may apply the results following 6.1.6, which dependupon that assumption.
Suppose first that Ca(Zs) ::::; M. Then (3) holds and we are in case (2) of
6.1.15, so (1) and (4) also hold. Therefore as (1) implies (2), Theorem 6.1.27 holds
in this case.
Therefore we may assume that Co(Zs) i M, so that Hypothesis 6.1.16 is sat-
isfied. Thus we can apply the lemmas in this subsection, which assume Hypothesis
6.1.16. We will derive a contradiction to complete the proof of the Theorem.
First n = 2 by 6.1.25.1, so IU: Cu(V)I = 4 by 6.1.20.4. Then by 6.1.21.4 and
6.1.25.3, ICu(V)/ El = 4. Finally Vis of order 16, and IE: VI ::::; 4 by 6.1.26.2, so
we conclude IUI ::::; 45. Hence m(U) ::::; 8.
Let W be an noncentral chief factor for Kon U. By 6.1.25.2, for each extension
field F of F 2 , the minimal dimension of a faithful FK*-module is (p-1)/2. Hence
as m(U) ::::; 8, p ::::; 17, sop = 11 or 13 by 6.1.25.2. But then p - 1 is the minimal
dimension of a nontrivial F2Zp-module, so we have a contradiction to m(U) ::::; 8.
This contradiction completes the proof of Theorem 6.1.27. D
6.2. Identifying M 22 via L 2 (4) on the natural module
In this section, we complete the treatment of groups satisfying Hypothesis 6.1.1,
by showing in Theorem 6.2.19 that M22 is the only group satisfying the conditions
established in Theorem 6.1.27. Then applying results in chapter 5, the treatment
of those groups containing a T-invariant L E Cj(G, T) with L/0 2 (L) ~ L 2 (2n) is
reduced in Theorem 6.2.20 to the case where n = 2 and Vis the sum of at most
two orthogonal modules for L/0 2 (L) regarded as n4(2). We treat that final case
in Part F2, which is devoted to the groups containing LE Cj(G, T) with L/02(L)
a group over F2.
So in this section, we continue to assume Hypothesis 6.1.1, and as in section
6.1, we let Zs:= Cv(T n L), VL := [V, L], and S := CT(Zs). As usual, Z denotes
D 1 (Z(T)). By Theorem 6.1.27, n = 2, and by 6.1.6, Cz(L) = 1 and VL is the
natural module for L/0 2 (L) ~ Lz(4). Applying these observations to R 2 (LT) in