12.8. GENERAL TECHNIQUES FOR Ln(2) ON THE NATURAL MODULE 849I*= Ki~ L2(4). Indeed ID: CD(A)I = 4, so
IB: Cs(I)I = IB: Cs(Ak)I:::; ID: CD(Ak)I = 4.
So as m(B) = 4, Cs(I)-/:-l. But this contradicts 12.7.22.5, since I is not a 2-group.
This final contradiction completes the proof of Theorem 12. 7.1.12.8. General techniques for Ln(2) on the natural module
When Hypothesis 12.2.3 holds and L ~ Ln(2) for n = 3, 4, 5, Theorems 12.4.2.1,
12.6.34, and 12.5.1 tell us that V is the natural module for L. We will encounter
a similar setup involving L 2 (2) after completing our treatment of the Fundamental
Setup. Thus in this section we establish some general techniques for treating all
four case simultaneously.
The hypotheses below reflect one difference between the treatments for n = 2
and n > 2: For n > 2, we have already analyzed the case where V is a TI-set in
Gin Theorem 12.2.13, so we simply exclude the groups appearing in conclusions
(2)-(4) of 12.2.13 as part of the operating hypothesis 12.8.1 of this section; then
by 12.2.13, Cc(Z n V) i. Mas Lis transitive on V#. However, the treatment of
the case where n = 2 and Vis a TI-set in G does not appear until the end of the
analysis of that case, so for the moment we instead assume Z:::; V and Ca(Z) i. M
as part of our operating hypothesis when n = 2.
Thus in this section, we assume the following hypothesis:HYPOTHESIS 12.8.1. Either (1) or (2) holds:
(1) Hypothesis 12.2.3 holds, with L/0 2 (L) ~ Ln(2), n = 3, 4, 5, and V the
natural module for L/0 2 (L). Further G is not Ln+1(2), Ag, or M24·
(2) G is a simple QTKE-group, T E Syb(G), Z := ~h(Z(T)), M E M(T),V := (ZM) is of rank 2, L = 02 (L) ~ M with M = !M(LT), CLT(V) = 02(LT),
and LT/0 2 (LT) ~ L 2 (2) ~ 83. Furthermore assume Cc(Z) i. M.
We adopt the following notation, which is consistent with that in Notation
12.2.5 when n > 2:
NOTATION 12.8.2. (1) Z := D 1 (Z(T)), M := Nc(L), Mv := NM(V), and
Mv := Mv /CM(V).
(2) n := m 2 (V), and for 1 :::; i :::; n, let Vi denote the i-dimensional subspace
of V invariant under T, Gi := Nc(Vi), and Mi:= NM(Vi). Let Li:= 02 (NL(Vi)),
unless n = 5 and i = 2 or 3, where Li:= NL(Vi)^00 • Set Ri := 02(LiT).
(3) Let z be the generator of V1 and G1 := Gi/V1. Set
Hz:= {HE H(L1T): H:::; G1 and Hf:. M}.
For HE Hz, set UH:= (VH), QH := 02(H), and H* := H/QH.
Note when n = 2 that V ~ M, so that Mi:::; Mv, and Li= 1. When n > 2,
V is a TI-subgroup in M and Mi :::; Mv by 12.2.6.
Lf'.
12.8.1. General preliminary results.
LEMMA 12.8.3. (1) Mv = GL(V), and eith~r Mv = L, or n =:= 2 and Mv =(2) L is transitive on i-dimensional subspaces of V, for each i.
(3) Gi is transitive on {Vg: Vi-:::; Vg}.
(4) G1 i. M.