3.1. COMMON NORMAL SUBGROUPS, AND THE qrc-LEMMA FOR QTKE-GROUPS 573HYPOTHESIS 3.1.5. T:::; Mo:::; M, HE H*(T,M), and VE R 2 (M 0 ) such that
R := 02(Mo) = Cr(V). Further either
(I) H n M:::; Na(0^2 (M 0 )), or
(II) H n M:::; Na(V).Observe that Hypothesis 3.1.5 includes the hypotheses of Theorem 3.1.1, other
than the condition that RE Syl 2 (0^2 (H)R): For example, Tis in a unique maximal
subgroup of H by 3.1.3.1.
The next result is a corollary to Stellmacher's qrc-lemma D.1.5 using Theorem
3.1.1.THEOREM 3.1.6. Assume Hypothesis 3.1.5. Then one of the following holds:
(1) There exists 1-=/:-Ro:::; R such that Ro ::::1 (M 0 ,H).
(2) Vi 02(H) and q(Mo/CM 0 (V), V):::; 2. If in addition Vis a TI-set under
M, then in fact q(Mo/CM 0 (V), V) < 2.
(3) q(Mo/CM 0 (V), V):::; 2.
PROOF. Assume that conclusion (1) does not hold. We verify Hypothesis D.1.1,
with Mo, Hin the roles of "G1, G2": By Hypothesis 3.1.5, T lies in both Mo and
H~so it is Sylow in both, since it is Sylow in G. By 3.1.5, V E R 2 (M 0 ) and
H E H*(T, M), so that H n M is the unique maximal overgroup of Tin H by
3.1.3.1, giving (1) of D.1.1. By 3.1.5, R = 02 (Mo) = Cr(V), which is (2) of D.1.1.Finally, our assumption that (1) fails is (3) of D.1.1. Thus we may apply the qrc-
Lemma D.1.5, to see (on combining its conclusions (2) and (4) in conclusion (ii)below) that one of the following holds:
(i) Vi 02(H).
(ii) q(Mo/CMo (V), V) :::; 2.
(iii) Vis a dual FF-module.
(iv) Rn 02 (H) ::::1 H, and U := (VH) is elementary abelian.Observe in case (ii) that conclusion (3) of Theorem 3.1.6 holds, so we may assume
that (ii) fails, and it remains to treat cases (i), (iii), and (iv).
Suppose case (iii) holds and let V* be the dual of V as an M 0 -module. Then
V is a faithful Frmodule for AutM 0 (V) ~ AutM 0 (V), so 02(AutM 0 (V*)) = 1
since V E R 2 (M 0 ). As (iii) holds, J := J(AutM 0 (V), V*) -=/:-1. Also Mo is an
SQTK-group using our QTKE-hypothesis, and hence so is the preimage in Mo ofJ. Therefore Hypothesis B.5.3 is satisfied with J, V* in the role of "G, V", so we
may apply B.5.13 to see that conclusion (3) again holds, completing the treatment
of case (iii).
As we are assuming that (ii) fails, q(Mo/CM 0 (V), V) > 1, so we may apply
D.1.2. By (2) and (3) of D.1.2,J(T) = J(R) i 02(H).
By (4) of D.1.2, His a minimal parabolic.in the sense of Definition B.6.1, and is
described in B.6.8.
In case (i), we argue that conclusion (2) holds: We will apply E.2.13, so we
need to verify that Hypothesis E.2.8 is satisfied with H n M in the role of "M",
and that F*(H) = 02 (H). We just saw that H is a minimal parabolic in the
sense of Definition B.6.1, and is described in B.6.8. As H E H(T), using our
QTKE-hypothesis and 1.1.4.6, His an SQTK-group with F*(H) = 02 (H). By