1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

12.S. GENERAL TECHNIQUES FOR Ln(2) ON THE NATURAL MODULE S55

Let F := Cu-(X) and suppose F > \/2. Then by (4), Fi. E, while by 12.8.9.5,

Cu(U^9 /Ziji) = E, so Fi. Cu(U^9 /Ziji). Now by 12.8.9.2, F::::; 02 (I 2 ) ::::; Na(X),

so [X, F] ::::; Zu n W^9 ::::; zg Vi by 12.8.10.1. Hence conjugating in h Xo :=

C_x(W /V2) =/= 1. If 1 =/= x E Xo centralizes W, then xis a transvection on U with

axis Wand center V2, so (6) holds. If x does not centralize W, then V 2 = [W, x] ::::;

[U, x] so as W is a hyerplane of U, m([U, x]) = 2 and again (6) holds. Thus (6) is

established. D

We are in a position to appeal to results in Volume I on centralizers with a

large almost extraspecial subgroup:

LEMMA 12.8.12. (1) Hypothesis G.10.1 is satisfied with if, u, V2, E, x, zg
in the roles of "G, V, Vi, W, X, Xo"·
(2) Let H2 := H n G2. Then X and Zij are normal in if 2 , so in particular
x :si 'i'.
(3) Hypothesis G.11.1 is satisfied.
(4) if and its action on U satisfy one of the conclusions of Theorem G.11.2.

PROOF. By 12.8.8.2, U is a symplectic space and if ::::; Sp(U), so Hypothesis
ft ft ftH
G.10.1.1 holds. By 12.8.11.2, E is totally isotropic. By 12.8.8.6, U = (V 2 ). As
T acts on Vi, T fixes the point V 2 of U, so part (a) of Hypothesis G.10.1.2 holds.
By 12.8.9.5, Eis the kernel of the action of X on U. Observe also that W = Vl-;
thus if x Ex -zg, then x ti-ZijE, so 12.8.9.6 shows that Cu-(x) ::::; w = V2..L,
establishing hypothesis (d). Hypothesis (b) follows from 12.8.11.5, hypothesis (c)
from 12.8.11.1, and hypothesis (e) from 12.8.11.3. This completes the proof of (1).

Next as [Vi, U] = Vi, H2 = Ca(Vi)U; so to prove (2), it suffices to show that

X and Zij are normal in Ca(Vi). But this follows as Ca(Vi) acts o~ UB and Vi.

0 bserve that part ( 4) of Hypothesis G .11.1 follows from ( 2), and hypothesis ( 3)

follows from 12.8.11.6. Thus (3) holds. Finally if is a quotient of the SQTK-group

H, so (3) and Theorem G.11.2 imply (4). D

Using Hypothesis 12.8.1, we can refine some of the results from sections G.10

and G.11:

LEMMA 12.8.13. (1) V::::; E.
(2) zg centralizes v.
(3) If n = 2, then Zu n Ziji = Z(h) = 1, so Zij ~ Zu and [Zu, Zij] = 1.
(4) If Zu >Vi then Zij =I= 1.
(5) If U is the 6-dimensional orthogonal module for F*(if) ~As, then 031 (H) =:
K E C(H) with K/02(K) ~ As, ZD := Zn Z(I2) =I= 1, VD := (Zfj) ::::; Zu,

VD E R2(KT), 1 =/= [ZD, K], and K = [K, Zij] E L1(G, T).

(6) Conclusion (4) of G.11.2 does not hold; that is, U is not the natural module

for F*(if) ~ A7..

(7) Conclusion {12) of G.11.2 does not hold.

(8) m3(CHCV2)) ::::; 1.

PROOF. As V::::; U and g E Na(V), V::::; UB, so (1) and (2) hold.

If n = 2, then by Hypothesis 12.8.1, Z 2 ~ Z ::::; V, so Z = Vi i. Z(I 2 ), and
hence Z(I 2 ) = 1. Therefore [Zu,Zij] ::::; Zu n Zij = Z(I 2 ) = 1. It follows from