14.5. STARTING THE CASE (VG1) ABELIAN FOR L 3 (2) AND L 2 (2) 1017PROOF. By 14.5.11.2, H* is a quotient of H+, while by 14.5.12.2, H+ is de-
scribed in case (2) of Theorem F.6.18. Thus (X+, Y+) = 03 (H+) 9:! 31+^2 or E 9 ,
and T+ 9:! E4. Then 14.5.6 completes the proof. D
We are now in a position to obtain a contradiction, and hence establish Theorem14.5.3. Let B := Z(P). By 14.5.10, J 1 (H)-=/=-1, so as T 9:! E 4 , the hypothesis
of D.2.17 holds. Thus in view of 14.5.13, case (4) of D.2.17 holds, with [VH,P*] =
[VH,B*] of rank 6. Then Vi= [V2,X]::::; [VH,P*], so VH = (V 1 H)::::; [VH,P*] and
hence VH = [VH,P*].
2In particular, VH = Vx EB Vl' EB Vl' , where (y) = Y and Vx := CvH (X) is
of rank 2. Further CvH (T) = (w, z) where (w) = Cvx (T) and z := wwYwY
2- Thus
(z) = CvH(YT), so V1 = (z). On the other hand,
z E Vi= [V2,X]::::; [VH,X],
and x acts on vt' since vt = CvH (X*Yi) and X*Yi is contained in the abelian
2 2group X B. Therefore [V H, X] = Vl' EB Vl'. This is a contradiction as z ~ Vl' EB Vl'
but we saw z E [VH,X].
This contradiction completes the proof of Theorem 14.5.3.14.5.2. Further preliminaries for the case U abelian. Recall we have
adopted Notation 12.8.2, including: Vi= (z), and
'Hz:= {H::::; G1: LiT::::; H and Hi. M}.In the remainder of this section, H denotes a member of 'Hz.
In contrast to the case where (VG^1 ) was non-abelian, when (VG^1 ) is abelian
we work with members Hof 'Hz possibly smaller than G 1.
Recall UH= (VH), QH = 02(H), and {Ji= Gi/Vi.
LEMMA 14.5.14. (1) Hypothesis F.8.1 is satisfied in H.(2) Hypothesis F.9.8 is satisfied in H, with V in the role of "V+ ".
PROOF. In view of Hypothesis 14.5.1, this follows from the list of equivalences
in 12.8.6. D
By 14.5.14, we may appeal to the results of sections F.8 and F.9. ·
LEMMA 14.5.15. (1) UH::::; Z(QH)), and UH E ~(H).
( 2) UH is elementary abelian. ·
(3) Assume L/0 2 (L) 9:! L3(2), and Li :::::! H. Then UH is the direct sum of
isomorphic natural modules for Li/ 02 (Li) = Li/ C L 1 (UH) 9:! Z3.
(4) QH = CH(UH)·
PROOF. Parts (1) and (4) follow from 12.8.4, (2) follows from Hypothesis 14.5.1
and 12.8.6, and (3) follows from 12.8.5.1. DNOTATION 14.5.16. By 14.5.14, Hypotheses F.8.1 and F.9.8 are satisfied in H,
so we can form the coset geometry r with respect to LT and H. Let b := b(r, V),
and choose a geodesic'}'o,'}'1, ··.,'Yb=: 'Y
as in section F. 9. Define UH, U 'Y, DH, D"', etc., as in section F. 9; in particular set
Ai := vr, recalling bis odd by F.9.11.1.