806 12. LARGER GROUPS OVER F2 IN .Cj(G, T)
I :S: Na(R) :S: N 0 (Cr(L)) :S: M = !Jvl(LT), contrary to I 1:. M. Thus Cr(L) = 1,
so that R = U by (*). Recall that this completes the proof of (1), and shows that
U = Ca(U).
Set Gu := Na(U) and Gu := Gu/U = Auta(U). We showed that Y :s;J
Na(R) = Gu, so as Y centralizes V, we conclude that j = (V^1 ) centralizes Y.
Now Y ~ Ln-l (2) has two chief factors on U, both isomorphic to the natural
module Un V, while j ~ D2p; it follows that IY is irreducible on U. Then as
Endy-(UnV) = F2, Gu= YxC 0 u(Y) ~ Ln-1(2) XL2(2), so j = C 0 u(Y) ~ L2(2),
and U is the tensor product of the natural modules for the factors. In particular
p = 3. Then as m 3 (YI) :S: 2 as Gu is an SQTK-group, it also follows that n < 5.
This completes the proof of (2), and hence of 12.2.25, under the assumption that
n > 3.
We turn to the case n = 3. This time let GR := Na(R), LR := NL(R),
MR:= NM(R), and GR:= GR/R. Since R = Cr(U), R is Sylow in Ca(R), while
GR E He by 1.1.4.4.6; thus R = Ca(R). As n = 3, U is of rank 4; so as JR: UJ :S: 2,
R is of rank k := 4 or 5. Thus GR :S: GL(R) = GLk(2). Further j ~ D 2 p with
U = [R, P] of rank 4, sop = 3 or 5. As H acts on R and I, as R = U x CR(I)
by(), and as JCR(I)J :S: 2, H centralizes CR(I). Thus His faithfully embedded in
GL(U) ~ GL4(2), with D2p ~ j :s;J if. We conclude that if~ 83, Z 2 x 83, D10, or
8z(2). Hence Tis cyclic or a 4-group. On the other hand, LR~ Vx 83, so that Tis
noncyclic. Hence if~ Z 2 X 83 , and in particular p = 3. Furthermore LR :::;i MR, so
MR centralizes CR(LR) as JCR(LR)I :S: 2; hence MR is faithful on the complement
[R, LR] to CR(LR) in R in (). Next MR normalizes [Rn 02(L), LR] = V n U,
and hence normalizes Vas Vis a TI-set in G by 12.2.16. Therefore MR centralizes
Vas JVJ = 2. Thus 02 (LR) = 02 (MR) from the"structure of the normalizer of
02 (LR) in GL([R, LR]) ~ GL4(2), so that MR = LRT = LR. Next C 0 R (V) acts
on [R, VJ = Un V, so again as V is a TI-set in G, C 0 R(V) :S: MR, and hence
c 6 R(V) =LR~ Z2 x 83.
Let i E T-V; then 1 f. [UnV,i] :s: unv. But ifi = iJ9 for some g E GR, then
[R, i] = [R, v]^9 = (UnV)B, so as Va TI-set in G, we conclude V =VB, contradicting
v f. i. Therefore i^0 n V = 0, so by Burnside's Transfer Theorem 37.7 in (Asc86a],
G is 2-nilpotent. As Y = Co(GR)(V) ~ Z3 and P :S: O(GR), we conclude from the
structure of GLs(2) that GR~ 83 x 83 and Cr(L) = CR(Y) = CR(P) = Cr(I).
We saw earlier that To = Cr(L) n Cr(I) = 1, so we conclude Cr(L) = 1 and
R = U. As observed earlier, this establishes (1) and shows that U = Ca(U). Along
the way we established the other assertions of (2), and (1) implies (3). Thus the
proof of 12.2.25 is complete. D
By 12.2.25.2, Nyv(P) is a complement to U in NL(U). Further U is a homo-
geneous Cyr(P)-module, so there is a Cyr(P)-complement U 0 to V n U in U, and
hence UoCyr(P) is a complement to Vin NL(U). As NL(U) contains the Sylow