S76 3. DETERMINING THE CASES FOR LE .Lj (G, T)HE He, so UH E R2(H) by B.2.14. As Ki CH(UH), CH(UH) :::; kerMnH(H)
by B.6.8.6, and CH(U H) is 2-closed by B.6.8.5. So as J(T) is not normal in H,
J(T) i CH(UH)· Hence by E.2.3, K = K 1 · · ·K 8 , withs= 1 or 2, T permutes the
Ki transitively, Ki/CK 1 (UH) ~ L 2 (2n), A3, or As, S = Baum(T) = Baum(R) acts
on Ki, and either Sis Sylow in KiS, or [UH, Ki] is the As-module for Ki/02(Ki)·
In the latter case, by E.2.3.3, S is of index 2 in a Sylow 2-group Si of SKi and
Si :::; (SKinM;. Then by an argument near the end of the proof of 3.1.6, Si :::; R. So
in either case, Rn KE Syl 2 (K), and hence RE Syl 2 (KR). As we observed after
Hypotheses 3.1.5, this is sufficient to establish the hypotheses of Theorem 3.1.1.
Hence conclusion (2) holds by that result, completing the proof. D
Finally we extend Theorems 3.1.6 and 3.1.7, by bringing uniqueness subgroups
into the picture:
THEOREM 3.1.8. Assume Lo= 02 (L 0 ) :Si M with M = !M(L 0 T), and VE
R2(LoT) such that 02(LoT) = CT(V). Then
(1) q(LoT/CL 0 T(V), V):::; 2.
{2) Either
{i) q(LoT/CLoT(V), V):::; 2, or{ii) For each H E H*(T, M), V i 02(H). If in addition V is a TI-set
under M, then q(LoT/CLoT(V), V) < 2.
{3) Either:
{i) J(T) i CT(V), so Vis an FF-module for LoT/CLoT(V), or
{ii) J(T):::; CT(V), Z:::; Z(H) for each HE H*(T,M), and Z(L 0 T) = 1.
PROOF. Set Mo := LoT, and consider any HE H*(T, M). Observe that case
(I) of Hypothesis 3.1.5 holds. Further as M = !M(M 0 ) and Hi M, 02( (Mo, H}) =1. In particular, neither conclusion (1) of Theorem 3.1.6, nor conclusion (2) of 3.1.7
holds. Therefore since q(Aui£ 0 T(V), V) :::; q(AutLoT(V), V) from the definitions
B.1.1 and B.4.1, we conclude from Theorem 3.1.6 that conclusions (1) and (2) of
Theorem 3.1.8 hold.
If J(T) i CT(V), then conclusion (i) of (3) holds by B.2.7. On the other hand,
if J(T) :::; CT(V), then by the previous paragraph, conclusion (1) of 3.1.7 holds, so
conclusion (ii) of (3) is satisfied. D
In certain situations we will require a refinement of the qrc-Lemma making use
of information in D.1.3 and definition D.2.1.
LEMMA 3.1.9. Assume case {II) of Hypothesis 3.1.5 holds, with HE H*(T, M).
Further assume:
(a) q(Mo/CM 0 (V), V) = 2.
{b) M ~ !M(Mo).
{c) V:::; 02(H).
( d) V is not a dual FF-module for M 0.Set UH:= (VH} and Z := D 1 (Z(T)). Then UH is elementary abelian, and
{1) H has exactly two noncentral chief factors U1 and U 2 on UH.
{2) There exists A E A(T) = A(CT(V)) with Ai 02 (H), and for each su.ch Achosen with A02(H)/0 2 (H) minimal, A is quadratic on UH.
{3) For A as in (2), set B :=An 02(H). Then B = CA(Ui),
2m(A/B) = m(UH/CuH(A)) = 2m(B/CB(UH)),