3.1. COMMON NORMAL SUBGROUPS, AND THE qrc-LEMMA FOR QTKE-GROUPS 5772m(B/CB(Vh)) = m(Vh/Cvn(B))
for each h EH with [Vh,B] =f. 1, m(A/B) = m(Ui/Cu;(A)), and CuH(A)
CuH(B).
(4) Define
m := min{m(D): DE Q(AutM(V), V)}.
Then m(A/ B) ::;:: m.
(5) Assume 02 (CM(Z)) S:: CM(V). Then H/CH(Ui) ~ 83 , 83 wr Z 2 , 85 , or85 wr Z2, with Ui the direct sum of the natural modules [Ui, F], as F varies over
the 83-factors or 85-factors of H/CH(Ui)· Further J(H)CH(Ui)/CH(Ui) ~ 83,
83 x 83, 85, or 85 x 85, respectively.
(6) Assume that each {2, 3}'-subgroup of CM(Z) permuting with T centralizes
V, m :::0: 2, and each subgroup of order 3 in CM(Z) has at least three noncentral
chief factors on V. Then H/CH(Ui) ~ 83 wr Z 2 •
PROOF. Observe that hypothesis (a) implies:
(a') Vis not an FF-module for M 0.
We will first show that (a') and (b)-(d) lead to the hypotheses of the qrc-lemma
D.1.5.
Set R := CT(V). By (a'), J(T) s; CT(V) = R. Thus the hypothesis of Theorem
3.1.7 holds, and by B.2.3.3, J(T) = J(R).
Next by (b), there is no 1 =f. Ros; R with Ro :::::) (M 0 ,H). Thus conclusion
(1) of Theorem 3.1.7 holds, so that Z :S Z(H), and in particular H n M s;·ca(Z).
Further J(T) is not normal in H, so we conclude from 3.1.3.2 that His a minimal
parabolic in the sense of Definition B.6.1. Also (as at the start of the proof of
Theorem 3.1.6) Hypothesis D.1.1 holds with M 0 , Hin the roles of "G 1 , G 2 ". Thus
we can appeal to results in section D.l, and in particular to the qrc-lemma D.1.5.
Observe that (c) rules out conclusion (1) of D.1.5, and (a') and (d) rule out
conclusions (2) and (3), respectively. We rule out conclusion (5) of D.1.5 just as inthe proof of 3.1.6, using (c) to eliminate case (i) in that proof. Thus conclusion (4)
of D.1.5 holds, so UH is abelian, and H has more than one noncentral chi~f factor
on UH. This last condition together with (c) and (a') are the hypotheses of D.1.3.
Furthermore (a') gives the hypothesis of D.1.2, so by part (4) of that result, His a
minimal parabolic in the sense of Definition B.6.1, and is described in B.6.8.
Next (a) supplies the hypothesis of part (3) of D.1.3. Then (1) follows fromD.1.3.3. We saw earlier that J(T) = J(R), so by D.1.3.2 there is A E A(T) with
A 1:. 02 (H) and A quadratic on UH. Indeed from the proof ofD.1.3.2, our choice of
A E A(T)-A( 02 ( H)) with A0 2 ( H) / 02 ( H) minimal guarantees that A is quadratic
on UH, and that B :=An 02(H) = CA(Ui) for i = 1, 2. Thus (2) holds, and D.1.3
establishes the remaining assertions of (3).
By (3), m(Vh /Cvn(B)) = 2m(B/CB(Vh)), and Bis quadratic on Vh by (2),
so AutB(Vh) E Q(AutMn(Vh), Vh) by (a). Thus
m S:: m(B/CB(Vh)) S:: m(B/CB(UH)) = m(A/B),
establishing (4).
Set H := H/CH(Ui)· As His irreducible on Ui, 02(H) = 1, so Ui E R2(H).
As B = CA(Ui) and m(A/B) = m(Ui/Cu;(A)) by (3), A~ A/Bis anFF-offender
on ui. Therefore by B.6.9, H =YT where y := J(H, V), Y =Yi x ... x YB*, and