CHAPTER 13
Mid-size groups over F 2
In this chapter we consider the cases remaining in the Fundamental Setup
(3.2.1) after the work of the previous chapter. We make more use of the generic
methods for the F2 case, such as results from sections F.7, F.8, and F.9.
In Hypothesis 13.1.1, we essentially extend Hypothesis 12.2.3 which began the
previous chapter, by adding the assumption that G is not one of the groups which
arose in the course of that chapter. Then after some reductions in the initial sections
13.1 and 13.2, in the remainder of the chapter we assume an additional refinement
in Hypothesis 13.3.1.In particular in 13.1.2.3, we observe that the remaining possibilities for the
section L/02(L), with L E .Cj(G, T) in the FSU, are A5, L3(2), A5, A 6 , and
U3(3) ~ G2(2)'. The main goal of the chapter is to treat the latter three groups,thus reducing the FSU to the case where L/02(L) is L3(2) or A5·
In the natural logical sequence, the smallest simple group A 5 is treated last;
thus at that point, all other groups are eliminated, so that K/0 2 (K) ~ A5 for all
K E .Cj(G, T). However, to avoid repeating arguments common to both A5 and
A 6 , we prove such results simultaneously for both in sections 13.5 and 13.6. To do
so, we assume in part (4) of Hypothesis 13.3.1 (and similarly in the hypothesis of
13.2.7) that K/02(K) ~ A 5 for all KE .Cj(G, T), when the subgroup LE .Cj(G, T)
we've chosen satisfies L/0 2 (L) ~ A 5. That is, we don't make this choice until we
are forced to do so, after the treatment of the other groups.13.1. Eliminating LE .Cf(G, T) with L/0 2 (L) not quasisimple
We now state the initial hypothesis for the chapter, which excludes the groups
in the Main Theorem that have arisen so far under the FSU. Namely throughout
this section, we assume:
HYPOTHESIS 13.1.1. {1) G is a simple QTKE-group and TE Syb(G).
{2) G is not a group of Lie type of Lie rank 2 over F2n, n > 1.
(3) G is not L4(2), L5(2), Ag, M22 1 M23, M24 1 He, or J4.As usual let Z := fh(Z(T)).
As mentioned earlier, Hypothesis 13.1.1 essentially contains Hypothesis 12.2.3,aside from the assumption in Hypothesis 12.2.1 that there is some L E .Cj(G, T)
with L/0 2 (L) quasisimple. In Theorem 13.1.7, we show for each K E .Cj(G, T)
that K/02(K) is quasisimple.
We record some elementary consequences of Hypothesis 13.1.1.
LEMMA 13.1.2. Assume there is LE .Cj(G, T) with L/0 2 (L) quasisimple and
set M := Na(L). Then L is T-invariant, there exists a T-invariant member V of
Irr +(L, R2(LT)), and:
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