1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

3.3. NORMALIZERS OF UNIQUENESS GROUPS CONTAIN Na(T) 601

LEMMA 3.3.28. If L is L3(2) or U3(3), then QL = V x To and <P(QL) = 1.

Indeed if L is U3(3), then To= I and QL = V.

PROOF. Assume that L is L3(2) or U3(3). and set To := CT(L). By 3.3.26,

there is d ED - M with Vd i. QL.

Suppose first that L 3:! L3(2). As case (I) of 3.3.23 holds, D acts on the

preimage To in T of Z(T). Then as !Toi= 2, To= VdQL and m(To/CT 0 (V)) =I,

so m(QL/CQL(Vd)) =I= m(V/Cv(Vd)), and hence QL = VCQL(Vd). Now if

QL/CT(L) is the unique nonsplit extension of V with a I-dimensional submodule

described in B.4.8, then the fixed points of T 0 are contained in VCT(L), contrary

to QL = VCQL(Vd) with To= ifd. Therefore QL = V x To, so as <P(To) =I by

3.3.27.2, the lemma holds in this case.

Thus we may assume L 3:! U 3 (3). Notice that if To= I, then V = 02 (LT) by

3.3.25.3, so that the lemma holds. Therefore we may assume that To -=f=. I, and it

remains to derive a contradiction.

Set X := 02 (CL(Z)) and R := 02 (XT). By 3.3.25.2, D acts on Rand X.

Then Vd is elementary abelian and normal in the parabolic subgroup XT, so using

B.4.6, m(Vd) = 2 or 3, and hence m(V/Cv(Vd)) = 3. Then by symmetry between
V and Vd, m(Vd/Cvd(V)) = 3. Thus m(Vd) = 3 so as ifd '.'9 X, ifd = C.R(ifd)
is the unique FF-offender on Vin R by B.4.6.I3. Therefore CR(Vd) :::; VdQL, so
CR(Vd) = VdCQL(Vd). Also ICR(Vd)I = ICR(V)I = IQLI, so IQL : CQL(Vd)I =
!Vdl. Then as !Vdl = IV : Cv(Vd)I, QL = CQL(Vd)V. However in the unique

nonsplit extension of V/Cv(L) over a I-dimensional submodule described in B.4.6,

the fixed points of ifd are contained in V/Cv(L). Thus as QL = VCQL(Vd),
QL = VTo. Then since <P(To) =I by 3.3.27.2, <P(QL) = 1.
Again by B.4.6.I3, ifd is the unique member of P(R, V), and Cv(Vd) = Cv(a)

for each a E ifd - L. Therefore as QL = VCT(Vd) and m(Vd) = m(V/Cv(Vd)),

B.2.2I applied with QL in the role of "V" says Qt is the unique member of A(R)

with [QL, Qt] -=f=. I, so A(R) is of order 2. Then as D of odd order acts on R, D

normalizes QL, contrary to 3.3.6.b. This completes the proof. D

LEMMA 3.3.29. L is not L3(2).
PROOF. Assume L is L 3 (2). By 3.3.23.I, D acts on the preimage To in T of
Z(T). Thus as D i. M by 3.3.6.a and M = !M(LT), no D-invariant subgroup of
To is normal in LT. Hence J(T 0 ) i. QL by B.2.3.3, so there is A E A(To) with
A i. Q£. Then as !Toi = 2, To = \a)QL for a E A - QL. Now <P(QL) = I by
3.3.28, so CQL (A) = CQL (a). Therefore by B.2.2I, A(To) ={A, QL} is of order 2.
Thus as D is of odd order, D acts on Q L, so that D :::; M = !M (LT), contrary to
Df:.M. D

LEMMA 3.3.30. L is an A5-block.

PROOF. Assume otherwise. Then by 3.3.25.I and 3.3.29, L is a G2(2)-block,
and it remains to derive a contradiction. By 3.3.28, To = 1 and V = QL, while
by 3.3.7.2, V is an FF-module for LT, so V 3:! E54 is the natural module for
LT/V 3:! G2(2).
Define A1 as in B.4.6. Then by B.4.6, m(A1) = 3, P(LT, V) = Af, and

Cv(A 1 ) = Cv(a) is of rank 3 for each a E A1 - L. Let Ao be the preimage in

M of A 1 ; by B.2.2I there is a unique member A of A(Ao) with image A1. Hence
A(Ao) = {V,A}. By Burnside's Fusion Lemma A.1.35, Nu.(T) = f' is transitive