522 2. CLASSIFYING THE GROUPS WITH IM(T)I = 1
CH(Q):::; Q, so Q E Si,(G) by 1.1.4.3. In particular applying this observation to U
in the role of "H", U = 02 (U) E Si(G), completing the proof of (3). D
We now use our assumption that 8* -:f 0 to verify that /3 -:f 0:
LEMMA 2.3.3. Let DE 8* and SE Syl 2 (NM(D)). Then:
(1) U E /3 for each U in S2(M) with D < U.
(2) D < S, ISi <!Mb SE /3, and SE Syl2(Nc(D)).PROOF. To prove (1), assume D < U E S 2 (M). Then U satisfies (/30) in
Notation 2.3.1. By definition of D E 8 in Notation 2.2.4, D E Si(G), so also
U E Si(G) by 1.1.4.1; hence by maximality of D, U rf-8, so that Nc(U) :::; M.
Applying this observation to any W E S 2 (M) containing U, we obtain (/31) for
U. Next set E := 02 (Nc(D)). If D < E, then Nc(D) :::; Nc(E) :::; M by the
observation, contradicting DE 8. Thus D = 02 (Nc(D)). We saw DE Si(G), so
that Nc(D) E He, and hence Co 2 (M)(D):::; CNa(D)(D):::; D. Thus (/32) holds for
D, and hence also for the 2-overgroup U. This completes the proof that U E /3,
giving (1).
Next let S E Syl 2 (NM(D)); we may assume S :::; T, and hence S = Nr(D).
As D E 8, S :::; Nc(D) i M = !M(T), so S < T. In particular D < T, so
D < Nr(D) = S. Then S E /3 by (1), and hence S E Syb(Nc(D)) by 2.3.2.2,
completing the proof of (2). D
We now introduce further notation suggested by the GLS proof of the Global
C(G,T)-Theorem, in as yet unpublished notes slated to appear in the GLS series;
an outline of their proof appears in Sec 2.10 of [GLS94].
NOTATION 2.3.4. Let U(G) = UM(G) denote the set of pairs (U,Hu) such
that U E /3 and Hu E He(U, M). Write U =UM for the set of U E /3 such that
He(U, M) -:f 0. For H E 7-l, let U(H) = UM(H) consist of those (U, Hu) E U( G)
such that Hu :::; H.
Recall that there exists D E 8*. By 2.3.3.2, a Sylow 2-group S of NM(D)
is in /3, so Nc(D) E He(s, M) by the definition of 8 in Notation 2.2.4. Thus
(S, Nc(D)) E U(G) and SEU, so that
U(G) and U are nonempty,
and by 2.3.3, S E Syb(Nc(D)) and Nc(S) :::; M. Observe that if H, H 1 E H with
H:::; Hi then U(H) S U(H1).
NOTATION 2.3.5. Let I'= rM be the set of all HE 1-{ such that U(H) -:f 0. Let
I'* =I'M-consist of those H E r such that U(H) contains some member (U, Hu)
with U of maximal order among members ofU, and subject to that constraint, with
IHl2 maximal. Let r = r M consist of those H E r such that IHl2 is maximal
among members of r. Finally let I'o = I'o,M :=I' u·r "
If DE 8 and SE Syb(NM(D), then we saw a moment ago that (S, Nc(D)) E
U(Nc(D)), so that Nc(D) Er and hence r -:f 0. As r is nonempty, also I' and
r * are nonempty.
Observe that by that by 2.3.2.2, IHl2 = IH n Ml2 for each H E r, so the
constraints on the maximality of IHl2 amount to constraints on IH n Ml 2.
LEMMA 2.3.6. If HE ro, then IHl2;::: IVI for any v Eu.