1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

854 i2. LARGER GROUPS OVER F2 IN .Cj(G, T)

Next [Ziji, W] :::; ZijinU:::; Zu Vi by (1) and symmetry between U and UB. Thus

any x E zg either centralizes the hyperplane W of tr, or induces a transvection on

W with center V 2 , so (4) follows.

To prove (5), assume Ziji:::; U. Then by (1) and symmetry between U and UB,

zg = (Zu n zg )Vf, so Ziji = Z(h) x Vf by (2). Then as U is conjugate to U^9 in

h, the first assertion of (5) holds.

Next let Pi, ... , Pn-i denote the minimal parabolics of L with the usual or-

dering so that NL(Vi) = (Pj : j f. i). Define Hi := (0^2 (PJ) : j :::; i). We argue by

induction on j that each Hj centralizes Z(I 2 ); and then in particular Hn-i = L

centralizes Z(h), which will complete the proof of (5). First Hi= 02 (P) = 02 (h)

from 12.8.9.1, and hence Hi centralizes Z(h). Now suppose that [Z(h), Hj] = 1 for

some 1:::; j < n-1. Then HjT is a maximal parabolic subgroup of Hj+lT, and so

there is k E PJ+i -HjT such that Hj+l = (Hj, Hj) centralizes F := Z(I 2 )nZ(I 2 )k.

Now k E PJ+i :::; NL(Vi) :::; H:::; Na(U), so that Z(I 2 ) and Z(I 2 )k are hyperplanes

of Zu using the result of the previous paragraph. Hence FVi is of codimension

at most 1 in Zu and is centralized by HJ+i, so HJ+i = 02 (HJ+i) centralizes

Zu ;:::: Z(I 2 ) by Coprime Action. This completes our inductive proof of the remain-

ing assertion of (5).

Finally by 12.8.9.5, Czi (tr) = Czi (U) :::; zg n U, and the reverse inclusion is

immediate. Further C zi (U) = Ziji n QH by 12.8.4.4, and the remaining equalities

in (6) follow from (1) and (2). D

LEMMA 12.8.11. (1) [W,X]:::; E.

(2) if>(E) = 1, so E is totally isotropic in the symplectic space tr.

(3) X induces the full group of transvections on E with center V 2.

(4) CE(X) = V2.

(5) m(E) + m(X/Ziji) = m(tr) -1.

(6) If Co(X) > V2, then there exists 1 f. x E X such that m([tr, ±]) :::; 2 and

V-2:::; [U,±J.

PROOF. By 12.8.9.2, (1) holds. As E = W n X, [E, X] :::; Vf by 12.8.4.2. As

if>(E) :::; if>(U) n if>(U^9 ) = Vi n Vf = 1, (2) holds.

By 12.8.8.1, U = UoZu with Uo extraspecial. Let E 0 := EZiji n ug and V 0 :=

ViZiji n ug. Then EZiji = EoZiji and ViZiji = ViZiji = V 0 Ziji. As Vi.Ziji = VoZiji,

X = W^9 = Cug(Vi) = Cug(Vo). As Eis abelian and centralizes_ Ziji, E 0 is also

abelian. Therefore as Uo is extraspecial, we conclude from these two remarks that:

(!) X induces the full group of transvections on E 0 which have center Vf, and

centralize V 0.
Let e E E -V2. As EZiji = EoZiji, eZiji = e 0 Ziji for some e 0 E E 0. Now by
12.8.10.1, Zu n E:::; Zu n U^9 = (Zu n Ziji)Vi.:::; Zu n E, so that all inequalities are
equalities. Hence En ViZu = Vi(Zu n E) = Vi(Zu n Ziji), and so by symmetry
between U and U^9 , En ViZu = En ViZiji. Thus as e tj. V 2 , e tj. ViZiji, so as we
saw that V2Zt). = VoZiji, eo tj. VoZ&. Thus [e, X] = [e 0 , X] = Vi by (!). Hence (3)
holds, and of course (3) implies (4).
Next
m(U) = m(E) +m(W/E) + 1 = m(E) +m(X/EZiji) + 1 = m(E) +m(X/Ziji) + 1,

as E = Cx(U) by 12.8.9.5. That is, (5) holds.