CHAPTER 12
Larger groups over F 2 in .Cj(G, T)
In this chapter we consider the cases remaining in the Fundamental Setup
(3.2.1) after the work of the previous parts. Then we reduce that list further,
concentrating on cases which can be treated by methods such as weak closure andcontrol of centralizers of certain elements of V.
After an initial reduction in the first section 12.1, the cases that remain are
listed in part (3) of Theorem 12.2.2 in the second section. Then in Hypothesis12.2.3, we add the assumption that G is not one of the groups already treated in
earlier analysis; the latter groups are listed in conclusions (1) or (2) of Theorem12.2.2. In the remaining cases L/CL(V) is essentially a group defined over F 2.
Then the main goal of this chapter is to treat, and in most cases eliminate, thelargest of those groups over F2: namely A5, A1, £4(2), and £5(2).
12.1. A preliminary case: Eliminating Ln(2) on n E9 n*
In this section we complete our analysis of case 3.2.5.3 of the Fundamental
Setup (3.2.1), where Vis a sum of two T-conjugates of Vo E Irr +(L, R2(LT), T).
Recall that most such cases were eliminated in Theorem 7.0.1. Thus it remains to
consider the cases where L/CL(V) ~ £4(2) or £5(2), and Vo is a natural module
for L/CL(V). We eliminate these cases using the weak-closure techniques of part
3, together with reductions from chapters E.6 and 11. We must work a little harderhowever, because m(M/CM(V), V) = 2, so that Theorem E.6.3 is not available to
giVe an initial lower bound on r(G, V).
Once this case is eliminated, we will have completed the treatment of the cases
in the FSU where Lis T-invariant and Lis not irreducible on V/Cv(L); for recall
chapter 10 completed the treatment of the case where Lis not T-invariant, while
Theorems 6.2.20 and 7.0.1 treated the cases where Vis not an FF-module.Thus at the end of this section, the treatment of the FSU will be reduced to
the cases described in 3.2.8. The first four subcases of 3.2.8 include all cases whereL/CL(V) is defined over F2n with n > 1, and those cases were handled in Theorems
6.2.20 and 11.0.1. Hence after this section it remains only to treat the cases where
L/CL(V) is a group defined over F 2 ; by convention we include A 6 and A 7 among
such groups.
While in this section L/CL(V) is also a group over F 2 , the fact that Lis not
irreducible on V makes the treatment of this case easier, and different from the
treatment of the generic case of groups over F 2.
So in this section we assume G is a simple QTKE-group, T E Syh(G), L E
.C:j(G, T) with L/0 2 (£) ~ Ln(2), n = 4 or 5, M := Na(L), V E R 2 (M), M :=
M/CM(V) ~ Aut(Ln(2)), and V =Vi EEl 112, with Vi the natural module for Land
112 = Vl fort E T-LQ, where Q := 02(LT) = CT(V). Thus 112 is the dual of Vi as
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