1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

6.2. IDENTIFYING M22 VIA L2(4) ON THE NATURAL MODULE 689

LEMMA 6.2.16. {1) v n Zu = z, so dim(Vu) = 2.

(2) (J = (Z1/) and 02(H) = 1.

{3) K* ~k.

(4) Vu = [V, U], and V is generated by an involution in Sp(U) of type a 2 •

(5) CH(V) = NHCVu) = H n M.

{6) N iICVu) is not transitive on v!f.

PROOF. Part (1) follows from 6.2.15.2, and part (3) from 6.2.15.3. As U =

(Z1/), (J = (Z1/), so as Zs S Z(T), (J E R 2 (H) by B.2.13, establishing (2). By

6.2.10.2, [U, V] S Vu, so by 6.2.15.2, Vu= [U, V] is ofrank 2 and U = V9Cu(V) for
some g E H: Thus Vis generated by an involution of type a 2 or c 2 in Sp(U) in the
sense of Definition E.2.6. Indeed for y E V9 - Z and v E V - U, [y, v] E Cvu (y) as
y induces an involution on V, so (f), f)v) = 0 and hence iJ is of type a 2 , establishing
(4). As there is a unique involution i E Sp(U) of type a2 with [U, i] =Vu, it follows

that Nfl(Vu) = Cil(V).

Let h E CH(V); then Vh = V by (3), so that h acts on [U, V*] = Vu.

Thus CH (V) S NH (Vu). But by the previous paragraph, N iI CVu) = C iI (V),

so NH(Vu) = NH(Vu) = CH(V). Finally NH(Vu) S H n M by 6.2.5.6, while

H n M = NH(V) by 6.2.9, and NH(V) acts on V n U =Vu, so (5) holds.

By 6.2.9, H n M acts on Zs, so (5) implies (6). D

Let Ls := 02 (NL(Zs)), l E Ls - H, E := Un U^1 , W .-Cu(Zs), and
X := Cuz(Zs). Observe as Zs SU that Zs s U^1 , and hence

Z S Zs S E.

LEMMA 6.2.17. {1) Zu n Zh = 1.

(2) Zu n U^1 = (Zu n Zh)Z.

A AJ_ • A A

{3) W =Zs and [X, W] SE.

(4) E is totally singular.

(5) For x EX -Zh, Co(±) SW.

{6) Cx(U) = ECzz(U).

(7) X induces the full group of transvections on E with center Zs.
{8) CE:(X) =Zs.
{9) vs .X.
(10) m(E) + m(X/Zh) = m(U) -1.
PROOF. Part (1) follows as V n Zu = Z by 6.2.16.1..
Next (U^1 ) = Z^1 and X acts on Zu, so [Zu n U^1 ,X] S Zu n Z^1 = 1 by
(1). Thus Zu n U^1 S Z(X). By 6.2.15.1, U = UoZu with Uo extraspecial, so
ZhZ = Z(Cuz(Z)) = Z(X). Therefore Zu n U^1 S ZhZ, so as ZS Zu n U^1 , (2)
holds.
Observe Hypothesis G.2.1 is satisfied with Z, Zs, Ls, Hin the roles of "Vi, V,
L, H", and set I:= (U, U^1 ) and P := 02 (1). As U is nonabelian by 6.2.15.1, while
Ls/0 2 (Ls) ~ L 2 (2)', the hypotheses of G.2.3 are also satisfied. So by that lemma,
I= LsU, P = WX, 1 <Zs SES Pis an I-series such that [J,E] S Zs, and
for some nonnegative integer s, and P / E = W / E EB X / E is the sum of s natural
modules for I/P ~ L2(2) with W/E = CP/E(U). Now V = [V,Ls] S Ls SI, so