850 12. LARGER GROUPS OVER F2 IN .Cj (G, T)
PROOF. Part (1) is an immediate consequence of Hypothesis 12.8.l. Then (1)
implies (2), and (2) and A.l.7.1 imply (3).
Assume case (1) of Hypothesis 12.8.l holds. Then Hypothesis 12.2.3 holds,
but conclusions (2)-(4) of Theorem 12.2.13 are excluded by that hypothesis. Thus
conclusion (1) of Theorem 12.2.13 holds, so that Cc(v) i M, and then (4) follows
from the transitivity of L on nonzero vectors of V in (2). Finally when case (2) of
Hypothesis 12.8.1 holds, (4) is a consequence of the assumption in that hypothesis
that Cc(Z) i M. D
By 12.8.3.4, G1 E Hz, so Hz =f 0. Observe that Hz ~He by 1.1.4.6.
LEMMA 12.8.4. Let H E Hz. Then
(1) Hypothesis G.2.1 is satisfied.(2) fjH ~ 01(Z(QH)) and fjH E R2(fI).
(3) <P(UH) ~ V1.
(4) QH = CH(UH), so H* is the image of Hin GL(UH) under the represen-
tation of H on UH by conjugation.
PROOF. As £ 1 is irreducible on V, (1) holds. Then G.2.2 implies (2) and (3).
If (4) fails, then Y := 02 (CH(UH)) =J l. But by Coprime Action, Y ~ Cc(V) ~
Mv, so [Y,L] ~ CL(V) = 02(L). Hence L normalizes 02 (Y02(L)) = Y, so that
H ~ Nc(Y) ~ M = !M(LT), contrary to the choice of Hi M. DLEMMA 12.8.5. Assume n > 2, so that L1 =f l.
( 1) If H E Hz with L1 :":) H, then UH is the direct sum of copies of the natural
module V for Li ~ Ln-1 (2)'.(2) If L1 :":) G1, then for 1 < i < n, Gi ~ M and V is the unique member of
vc containing Vi, so that m(V n VB) ~ 1 for g E G - Mv.
PROOF. Observe Vis the natural module for Li/02(L1) ~ Ln-1(2)', so (1)
holds as UH= (VH). Now assume £ 1 :":) G 1. Then for 1 < i < n, NL(Vi) inducesGL(Vi) on Vi by 12.8.3.1, so that Gi = Cc(Vi)NL(Vi) and Cc(Vi) ~ G1 ~ Nc(L1).
Hence Gi acts on(Lf : g E Gi) = (Lf: g E NL(Vi)) = L.
So Gi ~ Nc(L) = M. Then Gi =Mi ~ Mv, so the remaining assertions of (2)
follow from 12.8.3.3. DThe next lemma 12.8.6 shows that the condition "UH is abelian for all H E Hz''
is equivalent to "(VG^1 ) abelian". Much of our remaining work on the F 2 -Case is
partitioned via the cases "UH abelian for all H E Hz" versus "(VG^1 ) nonabelian".
We will discuss this distinction further after 12.8.6.
LEMMA 12.8.6. The following are equivalent:
(1) UH is abelian for each HE Hz.
(2) (VG^1 ) is abelian.
(3) If g E G with v n VB =f 1, then [V, VB]= 1.(4) Hypothesis F.8.1 is satisfied for each H E Hz·
(5) Hypothesis F.9.8 is satisfied for each H E Hz, with V in the role of "V+ ".
PROOF. First (1) implies (2) trivially as G1 E Hz. By 12.8.3.3 and the tran-
sitivity of L on V#, (2) implies (3). If (3) holds, then condition (a) of Hypothesis