1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

69a 6. REDUCING L2(2n) TO n =^2 AND V ORTHOGONAL

By definition of the bilinear form on fJ, Z g is the image of Gu (Zs) = W in U,

and the image of a subgroup Y of U in fJ is totally singular iff Y is abelian. As P / E
!s abelian, [X, W] ::; E, completing the proof of (3). As if>(E) ::; if>(U) n if>(U^1 ) =
z n zl = 1, (4) holds.

Pick u E U - W; from the action of I on P/E, the map rp : X ---+ W/E

defined by rp(x) := [x, u]E is a surjective linear map with kernel E. In particular

as Z::; E, Gu(x)::; W for each x EX - E. Further setting D := rp-^1 (ZuE/E),
D = Gx(U/ZuE). As P/E is a sum of natural modules for I/P, DZu = (Zij)E =
ZuZhE, so D = ZhE. Thus Gx(U) :S: Gx(U/ZuE) = D = ZhE. In particular

for u EU - W, Gx(u) :S: Z^1 E, and hence (5) follows.

Let R := GT(U), and UR := Gu(R) with preimage UR. By a Frattini Argu-

ment, H = GH(U)NH(R), so as Zs :S: UR and U = (Zff), U = URZu. Therefore
as Zu::; W < U, R centralizes u EU - W. In particular Gx(U)::; Gx(u)::; Z^1 E,
so (6) holds.
Let Ea:= EZhnU6 and Za := ZZhnU6. Then EZh :S: U^1 = U6Zh, so EZh =
EaZh, and similarly ZsZh = ZZh = ZoZh. Thus X = Gui(Zs) = Gui(Za). As

EZh is abelian, so is Ea. Therefore as Ua is extraspecial, we conclude that:

X induces the full group of transvections on Ea with center Z^1 centralizing Za.

(!)

Let e E E - Zs. As EZh = EaZh, eZh = eaZh for some ea E Ea. By (2),

En ZsZu = Zs(E n Zu) = Zs(U^1 n Zu) = Zs(Zu n Zh) =En ZsZh.

Thus as e tj. Zs, e tj. ZsZh, so as ZsZh = ZaZh, ea tj. ZaZh. Thus [e, X]

[ea, X] = Z^1 by (!). Hence (7) holds and of course (7) implies (8). Finally

m(U) = m(E) +m(W/E) + 1 = m(E) +m(X/EZh) + 1 = m(E) +m(X/Zh) + 1,

where the last equality follows from (6). Thus (10) holds. D

LEMMA 6.2.18. (1) X and Zh are normal in GH(V).

(2) if and its action on fJ satisfy one of the conclusions of Theorem G.11.2.

PROOF. We first verify that fJ, if, Zs, E, X, Zh satisfy Hypothesis G.10.1 in

the roles of "V, G, V 1 , W, X, Xa". As if>(X) :S: Z^1 :S: U, Xis elementary abelian,

and E is totally singular by 6.2.17.4. By construction condition (a) of part (2) of

Hypothesis G.10.1 holds. Conditions (b), (c), (d), and (e) are parts (10), (3), (5),

and (7) of 6.2.17, respectively. So Hypothesis G.10.1 is indeed satisfied.

Let MH := H n M. By 6.2.16.5, MH = GH(V), and by 6.2.9, MH = NH(Zs),

so since [Zs, U] = Z, we conclude MH = UGH(Zs). Then as X and Zh are normal

in GH(Zs), (1) holds.

Next we verify Hypothesis G.11.1. Case (ii) of condition (3) of that Hypothesis

holds by 6.2.17.9 and 6.2.16.4. As MH contains the Sylow 2-subgroup T of iI, con-

dition (4) of Hypothesis G.11.1 follows from part (1) of this lemma. So Hypothesis

G.11.1 is verified. Then part (2) of the lemma follows from Theorem G.11.2. D

We can now complete the elimination of the case remaining after 6.2.13.

THEOREM 6.2.19. If G satisfies Hypothesis 6.1.1, then G ~ M 2 2.