CHAPTER 11
In this chapter, we complete the elimination of the groups possessing a pair L,
V arising in the Fundamental Setup (3.2.1) such that L/0 2 (L) is of Lie type of Lie
rank 2 over a field of order 2n, n > l.
Choose V so that L, V are in the FSU and L/02(L) is of Lie type of Lie rank
2 over a field of order q := 2n, n > l. By Theorem 7.0.1, V is an FF-module. The
weak closure parameters of FF-modules make it difficult to do weak closure without
first doing some extra work. Furthermore corresponding local configurations do
actually occur in suitable maximal parabolics in non-quasithin shadows given by
certain groups G of Lie type and Lie rank 3: namely for L ~ SL3(q), in G ~ L4(q),
Sp5(q), nt(q).2, and fi8(q); and for L ~ Sp4(q), in G ~ Sp5(q).
- We restrict attention at this point to q = 2n for n > 1, largely because for such
q, L has a Cartan subgroup X of p-rank 2 for primes p dividing q - l. Using our
quasithin hypothesis, G contains no member of H(X) of larger p-rank, whereas the
the groups of Lie type in the previous paragraph do contain such subgroups. This
leads to a contradiction, which does not arise in the shadows of groups over the
small field F 2 ; the more complicated treatment needed for the subcase of L of rank
2 over F 2 is postponed to part 5.
Thus in this chapter we will prove:THEOREM 11.0.1. Assume G is a simple QTKE-group, T E Syh(G), and
LE .Cj(G, T). Then L/0 2 (L) is not isomorphic to (S)L 3 (2n), Sp4(2n), or G2(2n)
with n > 1.
Throughout this chapter we assume Lis a counterexample to Theorem 11.0.1
By 1.2.1.3, Lis T-invariant, so by 3.2.3, M := Na(L) E M(T), M = !M(LT),and we can choose V so that L and V are in the FSU. In particular let VM :=
(VM), VM := VM/CvM(L), Mv := NM(V), and Mv :=:= Mv/CM(V). Let TL :=
T n L0 2 (LT) and let X be a Hall 2'-subgroup of NL(TL); since n > 1, mp(X) = 2
for each prime divisior p of IXI (see 11.0.4). As mentioned earlier, the Cartan
subgroup X will provide a main focus for our analysis. Set Z := !1 1 (Z(T)) and
abbreviate q := 2n.
Lemmas 11.0.2, 11.0.3, and 11.0.4 collect observations from various earlier re-
sults, and provide a starting point for the analysis.
LEMMA 11.0.2. (1) VE Irr+(L,R2(LT)) and Vis T-invariant. Moreover T
is trivial on the Dynkin diagram of L/02(L).
(2) V/Cv(L) is the natural module for L/02(L) ~ L ~ SL3(q), Sp4(q), or
G2(q).
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