i2.8. GENERAL TECHNIQUES FOR Ln(2) ON THE NATURAL MODULE 853(2) 02(I2) = WX, [E,h] =Vi, and 02(I2)/E = W/E tB X/E is the direct
sum of natural modules for h/02(I2) ~ L2(2) ~ 83.
(3) Co 2 (I 2 )/E(u) = [02(I2)/E, u] = [X/E, u] = W/E for u EU - W.
(4) For y Ex - w, Cu(Y) s w.(5) Cx(U) = Cx(U) = E.
(6) For u EU - W, Cx(u) s ZijE.
(1) V[ n Zu = 1.
PROOF. Observe (7) holds as g E NL(Vi) - Gi, and V n Zu =Vi by 12.8.8.3.
As Vis the natural module for LT and 02 (LT) = CLr(V), P = 02 (P)T withCp(V2) = 02(P) and P/02(P) = GL(Vi) ~ L2(2). As U is nonabelian, [Vi, U] -:f:. 1
by 12.8.8.6, so 02 (P) = [0^2 (P), U] and P = (U, Ug)0 2 (P). Thus I 2 = 02 (P)U.
As Autp(V2) = GL(Vi), G2 = Ca(Vi)P, so as Ca(V2) s Gi s Na(U), we conclude
I2 ~ (U, Ug) = (U^02 ) ~ G2, so 02 (P) = 02 (I 2 ) ~ G 2 , completing the proof of
(1).
By 12.8.8.6, Hypothesis G.2.1 is satisfied with 02 (P), Vi, 1 in the roles of "L,
V, Li"; further U = (V 2 H) by 12.8.8.6, so (2) and (3) follow from G.2.3.
Pick u EU - W; by (3), [X,u] s W, so we can define <p: X-+ W/E by
<p(x) := [x, u]E. Set D := <p-i(ZuE/E). By (3), Cx(U) s D, and
m(X/D) = m(W/ZuE).
As 02(h)/E is the sum of natural modules for h/02(I 2 ) ~ 83 by (2),DZu = (Z{J)E = ZuZijE,
so D = ZijE. Thus ify '/'. ZijE, then [y,u] ¢. ZuE, and in particular [y,u] tf-Zu,
so (6) holds. Similarly for y E X - W and u E U - W, [y, u] ¢. Eby (2) and in
particular [y, u] tf-Vi, so [y, u] #-1. Thus (4) holds.
Of course E s Cx(U) s Cx(U). Let R := Cr(U) and V 0 := Cu(R). By
a Frattini Argument, I{ = CH(U)NH(R); so as V 2 s Vo, as V 0 is normalized
by NH(R), and as (J = (f1 2 H), we conclude that U = VoZu. In particular as
Zu s W < U, R centralizes some u EU -W, so by (4), X n Rs X n W = E,
completing the proof of (5). D
LEMMA 12.8.10. (1) Zu n ug = (Zu n Zij)Vi.
(2) Zu n Zij = Z(h).
(3) If Zu n Ug >Vi, then Zn Z(h) #-1.
(4) [W,Zij] s ZuV2, so m([U,x]) s 2 for x E Zij.
(5) If Zij s U, then Zu = Z(h) x Vi and [L, Z(I2)] = 1.
(6) Czf; (U) = Czf; (U) = Zij n QH = Zij n U = (Zij n Zu )V[ = Z(h)V[ s U.PROOF. By 12.8.9.7, V{ n Zu = 1, and by 12.8.8.1, Z(W) = V 2 Zu. Thus by
symmetry between U and Ug, Vin Zij = 1 and Z(X) = V2Zij = ViZij.
By 12.8.4.2, [Zu n Ug,X] s V{ n Zu = 1, so Zu n Ug s Z(X). Therefore
Zu n Ug s ZijVi by the previous paragraph, so as Vi :S Zu n Ug, (1) holds by the
Dedekind Modular Law.
By 12.8.9, h = (U, Ug), so Zu n Zij s Z(I2). To prove the reverse inclusion,
observe by 12.8.9.2 that Z(I2) s W n X, so Z(I2) = Z(I2) n U :S Z(U), and
similarly Z(I 2 ) S Z(Ug). Thus (2) holds. As T acts on Vi, T acts on h by
12.8.9.1, and hence on Z(I2). Further if Zu n Ug > Vi then Z(I2) #- 1 by (1) and
(2), so Cz(r 2 )(T) #- 1 and hence (3) holds.