1020 i4. L 3 (2) IN THE FSU, AND L 2 (2) WHEN .Cr(G, T) IS EMPTYAssume case (2) of Hypothesis 14.3.1 holds. Then (3) follows from 14.2.2.4. As-sume [UH, QH] = l. Then by 14.5.15.4, QH = CH(UH). By 14.5.15.1, 02(H/QH) =
1, so that UH E R2(H). Suppose there exists K E C(H). As QH = CH(UH),
K E £1(G, T), contradicting 14.3.4.4. So H is solvable by 1.2.1.1, and henceO(H) -:/= l. Then UH = [UH,O(H)] EEl CuH(O(H*)) by Coprime Action. As
H > QH = CH(UH), [UH,O(H*)]-:/= 0. Then Zn [UH,O(H*)]-:/= 0, contradicting
H:::; Ca(Vi) since Z =Vi when L/0 2 (L) ~ L2(2)'. D14.6. Eliminating L 2 (2) when (VG^1 ) is abelian
In this section we assume Hypothesis 14.5.1 holds with L/0 2 (L) ~ L2(2)'; in
particular, U := (V^01 ) is abelian. Also Hypotheses 14.3.l.2 and 14.2.l are satisfied,
so we can appeal to results in sections 14.2 (with Lin the role of "Y"), 14.3, and
14.5.We will see in Theorem 14.6.25 that no further quasithin examples arise beyond
those which we characterized earlier in Theorems 14.2. 7 and 14.2.20, where U was
nonabelian. Thus in this section we will be working toward a contradiction. Indeed
as far as we can tell, there are no shadows.
As usual Z := fh(Z(T)) for TE Syh(G). Recall that by Hypothesis 14.2.1.4,
Vis of rank 2 with V :SJ M. Recall also that Cr(L) = 1by14.2.2.2.
We also adopt Notation 12.8.2: Thus Vi := Zn V = Z since Z is of order
2 by 14.2.2.6, and Gi = Na(Vi) = Ca(Z) = Mc E M(T). Recall also that
Li := 02 (CL(Vi)) ~ 1; this simplifies the application of results from sections 14.3
and 14.5 involving Li. For example as Li = 1, 14.2.5 says that:
H(T, M) = Hz.
For the remainder of this section, H denotes a member of H(T, M).
Recall Gi := Gi/Vi and notice fI makes sense as H :::; Gi by definition of
Hz. As U is elementary abelian and H:::; Gi, UH := (VH) :::; U is also elementary
abeliari (cf. 14.5.15.2).LEMMA 14.6.l. {1) Gi = !M(H).
{2) 02,p(H) n M = 02(H) for each odd prime p.
{3) If KE C(H), then Ki M.
(4) If 1 i= X = 02 (X) :SI H, then XT E H(T, M).
{5) 02,F•(H) centralizes fh(Z(0 2 (H))).(6) If 02(H) :S: Ti :SI T, then Na(Ti) :S: Na(fh(Z(Ti))) :::; Gi.
PROOF. Part (1) is 14.2.3, part (3) is 14.5.19, and part (2) follows as case (1)
of 14.5.20 holds because L/02(L) '¥-L 3 (2). Under ~he hypotheses of (4), 02 (X) <
02,F•(X), and 02,F•(X) i M by (2) and (3), so (4) holds.
Let R := 02(H), W := fh(Z(R)), and fI := H/CH(W). Suppose there is
K E C(H) with [W, K] i= l. Then as R. centralizes W, K E £ 1 ( G, T) by A.4.9,
contrary to 14.3.4.4. This contradiction shows that 02 ,E(H) centralizes W.
Suppose (5) fails. Then by the previous remark, for some odd prime p, X :=
02 (0 2 ,p(H)) is nontrivial on W. As 02 (X) :::; R:::; CH(W), Xis of odd order, so
W = [W, X] EElCw(X) by Coprime Action. Then as [W, X] i= 0, Z:::; [W, X] since Z
has order 2. However X:::; H:::; Gi by (1), so also Z :=:; Cw(X). This contradiction
establishes (5).