14.5. STARTING THE CASE (vGi) ABELIAN FOR Ls(2) AND L2(2) 1019Set Yz := 02 (02,z(Y)), and Ye := 02 (CyM(L/02(L)). By (!), Yz ej_ X, so
Yz i Ye. On the other hand if Ye= YM then YM EX, so Yz:::; Na(YM):::; M by
(!);then Yz:::; YM =Ye, contrary to the previous remark, so:
Ye< YM· (*)
It follows that p = 3: For if p > 3 then T permutes with no p-subgroup
of L/02(L), so that YM :::; Ye, contrary to (*). If L/02(L) ~ L 2 (2)', then YM ·
centralizes V /V1 and V1 of order 2, and hence centralizes V by Coprime Action, so
again YM centralizes L/02(L), contrary to (*). Therefore L/0 2 (L) ~ L 3 (2). Next
we claim:
(!!)
For if L1 i Y, then
[YM, T n L]:::; CL(V1) n YM = L102(L) n YM:::; 02(YM),
so YM centralizes (T n L)/02(L) and hence also L/0 2 (L) by the structure ofAut(L3(2)), again contrary to (*). We have established the first three statements
in (2), so it remains to show that one of cases (i)-(iii) holds.
If Y/02(Y) is cyclic then Y = Li by (!!) since Y/0 2 (Y) is of exponent 3, so
conclusion (i) of (2) holds. Therefore by A.1.25.1, we may assume Y/0 2 (Y) ~ E 9or 31+^2. In the latter case, Yz satisfies the hypotheses of "Y", so we conclude
L 1 = Yz from (!!). Thus in either case, L 1 is normal in Y.Let H* := H/QH. As M = LCM(L/02(L)) and L1 i_ Ye:
YM-=Li x Y 0. (**)
In particular if Y* ~ 31+^2 then Y i M by (*).
Next we claim that if Yi= 02 (Yi):::; Yis T-invariantwith Yi/0 2 (Yi) of order 3,
then Y1 :SM: For ifY1 i M, then as Na(T) :SM by 14.3.3.3, Y1T/02(Y1T) ~ 83.
Then as L 1 is normal in Y, the claim follows from 14.5.3.2. It then follows from
the claim that if Tacts reducibly on Y/0 2 ,w(Y), then Y:::; M. Now if Y ~ 3i+^2
we saw Y i M and L1 = Yz, so T acts irreducibly on Y* / LJ. and L1 = Y M, so
that conclusion (ii) of (2) holds.
Thus we may assume that Y* ~ E 9 • Then L 1 < Y so that Tacts reducibly
on Y, and hence Y :::; M by an earlier remark. Then Y = LJ. x Ye by (**),
with Ye of order 3. Then Ye E X, so Ye is not normal in H by (!). Therefore
as Aut(Y) ~ GL 2 (3) with Autr(Y) normalizing Ye, there is some 3-element
y E H - Y inducing an automorphism of order 3 on Y* centralizing LJ., with T
acting on Y+ := Y(y). As M = LCM(L/02(L)), Y+ i M, so Y+T E Hz, and
then we may assume H = Y+T. If y* has order 3, then Yf' ~ 31 +^2. As T is not
irreducible on Yf'/ L]', this is contrary to an earlier reduction. Hence y has order 9,
and we may choose y so that Yo := (y, L 1 ) :::1 H with Y 0 /02(Yo) ~ Z 9 , and thus
conclusion (iii) of (2) holds. D
LEMMA 14.5.21. (1) The map r.p defined from QH/CqH(UH) to the dual space
of UH/CuH(QH) by r.p: xCqH(UH) 1-+ CuH(x)/CuH(QH) is an H-isomorphism.
(2) [UH, QH] = V1.(3) CH(V2) acts on L2, and m3(CH(Vi)) :::; 1.
PROOF. Part (1) is F.9.7.
Assume case (1) of Hypothesis 14.3.1 holds. Then (2) follows from 13.3.14 and
14.5.15.1, while (3) follows from parts (1), (2), and (5) of 13.3.15.