3.3. NORMALIZERS OF UNIQUENESS GROUPS CONTAIN Na(T) 603LEMMA 3.3.32. (1) Either QL = V is irreducible, or QL ~ E 32 is the quotient
of the permutation module on n modulo (en), denoted by "U" in section B.3.(2) Lt~ 85.
(3) D acts on the preimage T 0 in T of A 2 := ((1,2)(3,4), (5,6)).
PROOF. As Tc = 1 by 3.3.31, (1) follows from C.1.13 and B.3.1. Let P 1 be the
stabilizer in LT of {5, 6}, and P 2 the stabilizer of the partition { {l, 2}, {3, 4}, {5, 6} };set Ri := 02(Pi), and Xi := 02 (Pi)· Then P1 and P 2 are the maximal parabolics
of LT over T, P 1 has two noncentral 2-chief factors, P 2 has three noncentral 2-
chief factors, and 02(X2) is nonabelian with Z(X 2 ) = 1. Then P 1 does not satisfy
conclusion (4) of 3.3.20 and P2 does not satisfy conclusion (5c) of 3.3.20, so D acts
on Ri and R2. Thus D acts on Ti:= Ri n R2.
If Lt ~ 86 , then To = Ti and the lemma holds, so we may assume Lt ~ A 6 • ·
Thus ti = ((3, 4)(5, 6)). But then P(t1, QL) is empty by B.3.4.1, so J(T1) ::::;CLr(QL) = QL. Then as QL is elementary abelian by (1), J(T1) = QL ::::] LT, and
hence D::::; Nc(QL) ::::; M, contrary to 3.3.6.a. Thus the lemma is established. D
We can now obtain a contradiction, and complete the proof of Theorem 3.3.1.
In view of 3.3.32.1, QL is either the natural module for L denoted by "U 0 "
in B.3.2, or the quotient denoted "U" of the permutation module. Define Ai :=
((5, 6)), and A 2 as in 3.3.32.2. By 3.3.32.3, D acts on the preimage T 0 of A2 in
T, and as D i M by 3.3.6.a, D acts on no nontrivial subgroup of T 0 normal in
LT. In particular J(To) i QL by B.2.3.3, so there is A E A(To) with Ai QL. By
B.3.2, A = Ai for i = 1 or 2. By inspection, CqL (A) = CqL (a) for some a E A,
so by B.2.21 there is at most one member of A(T 0 ) projecting on Ai; if such a
member exists, we denote it by Ai. Thus A(To) <:;;; {QL,A1,A2}. Therefore as D
acts on A(T 0 ) but not on QL, and Dis of odd order, DL is transitive on A(To) of
order 3. Further D is transitive on the 2-subsets of A(To). This is impossible as
IA1QLI < IA2QLI·