4.1. SOME GENERAL MACHINERY FOR PUSHING UP 607Set Mi := NM_ (R), and pick R+ E Syl2(GM(M+/02(M+)) so that Ri :=
NR+(R) E Syl2(CM 1 (M+/02(M+))). If R = R+, then (2) holds by 4.1.2, so wemay assume that R < R+, and hence R < Ri. We will verify the hypotheses of
4.1.3, with Mi, NM 1 (Ri) in the roles of "Io, Ii". First I :::; Mi, and 02 (Mi) =f.
1 =f. 02(NM 1 (Ri)), since 1 =f. 02(!) = R :"'.:: 02(Mi) n 02 (NM 1 (Ri)). By a Frattini
Argument,
Mi= NM 1 (Ri)GM 1 (M+/02(M+)).
Finally M+ acts on R+ by A.4.2.4, and h~nce M+ n Mi= NM+(R).:::'.: NM 1 (Ri),completing the verification of the hypotheses of 4.1.3. Thus NM 1 (Ri) E 'T/ by 4.1.3.
Also [NM+(R),Ri] :"'.:: 02(M+) n Mi:"'.:: Rias Ri E Syl2(CM 1 (M+/0 2 (M+))), so
InM+ :"'.::NM+ (R) :"'.::NM+ (Ri). By construction 02(!) = R < Ri :"'.:: 02 (NM 1 (Ri)),
so I ;S NM 1 (Ri). Therefore as IEμ* by hypothesis, M = !M(NM 1 (Ri)) by 4.1.3.2.Then as Mi :"'.:: Nc(R), (2) follows.
A similar argument shows RE Syl 2 (GHnM(H+/0 2 (H+))): Assume that
R <RH E Syb(GHnM(H+/02(H+))).As CM(M+/02(M+)) :"'.:: M, RH is also Sylow in CHnM (M+/02(M+)). Set
Hi:= NHnM_(R) and choose RH so that Ri := NRH(R) E Syl2(GH 1 (M+/02(M+)).
By a Frattini Argument, Hi= NH 1 (Ri)GH 1 (M+/02(M+)). By (3),M+ n H = H+02(M+ n H)::::: H+RH,
and by A.4.2.4, H+ acts on RH, so
M+ n Hi= NM+nH1 (R)::::: NH1 (Ri).
Hence applying 4.1.3.1 to Hi, NH 1 (Ri) in the roles of "Io, Ii", we conclude
NH 1 (Ri) E 'T/· By construction, Hi :"'.:: H f:. M, so 1i(NH 1 (Ri),M) =f. 0, and
hence NH 1 (Ri) E μ. Also by construction, 02(I) = R < Ri :"'.:: 02(NH 1 (Ri)) and
arguing as above, I :S NH 1 (Ri). This contradicts our hypothesis that I E μ*,
completing the proof that RE Syl2(GHnM(H+/02(H+))). Then (4) follows using
(2).
As H+ :::::) H n M, RE !32(H n M) by C.1.2.4. By (2), NH(R) ::::: H n M, so
RE J3 2 (H) by C.1.2.3. By C.2.1.2, both 02 (H) and 02 (HnM) lie in R :"'.:: HnM,so in fact 02 (H) :::; 02(H n M) :"'.:: R, completing the proof of (5).
Let H:::; Hi EM. Then Hi E 1i(I, M), so all results proved for H also apply
to Hi. In particular by (5), 02(Hi n M) :"'.:: R :"'.:: H n M, and hence 02(Hi n M) :"'.::
02 (H n M). Now if F*(Hi n M) = 02(Hi n M), then
GHnM(02(H n M))::::: CH1nM(02(Hi n M))::::: 02(Hi n M)::::: 02(H n M),
so (6) holds. That is, if (6) holds for Hi, then it also holds for H, so we may assume
H =Hi EM. Now Cc(02(H)) :"'.:: Nc(02(H)) = H, while 02(H) :"'.:: 02(H n M)
by (5). Thus Co 2 (M)(02(H n M)) :"'.:: CM(02(H)) :"'.:: H n M, so H n ME 1ie by
1.1.4.5, proving (6). This completes the proof of 4.1.4. D
LEMMA 4.1.5. Let R+ E Syh(GM(M+/0 2 (M+)), and assume
1 =f. V = [V, M+] :"'.:: Di(Z(R+)).
Suppose IEμ* and R := 02(!) :"'.:: R+· Then
(1) V :"'.:: Z(R).
(2) IJV = [Di(Z(R+)),M+J, then Nc(V) :"'.:: M.
(3) Let HE 1i(I, M), and set H+ := 02 (M+ n H). Then V = [V, H+]·