2 CHAPTE R 1. GENERAL LEMMAS
that
yhl
G P and by control of fusion there is h2 G M such that
xh2
=
yhl.
Then
ghih^1
G Cx(x) M because x G C, whence g £ M. This establishes (IB).
Likewise, there is clearly a close connection between these definitions and the
notion of control of p-fusion (see [IG; Section 16]). For one thing, in view of Sylow's
theorem, the definition of T means that for an element x G TP{M) to lie in F,
it is necessary and sufficient that any X-conjugate of x lying in M be already
M-conjugate to x. The following lemma is therefore immediate:
LEMMA
1.2. Suppose that M controls X-fusion in some Sylow p-subgroup of
M. ThenC(X,M,p)=U{X,M,p).
Conversely (see Lemma 1.7 below), if U(X, M,p) contains as much as a single
p-central element of X, then M controls p-fusion in X.
By analogy with (IB), we also make the following definition:
DEFINITION 1.3. For any integer k 1, the set Uk = Uk(X, M,p) is defined as
Uk = {E\EM, E^Epkj and for any g G X, E9 M if and only if g G M}.
Again we suppress X, M and p when it is clear what they are.
We now fix X, M and p, use the notation of (1A), and establish a few key
elementary properties of these sets of uniqueness elements. First, it is immediate
from the definitions that
(1C) C, U and Uk are invariant under M-conjugation.
LEMMA
1.4. IfYM and Y contains an element of U or Uk for some k,
then NX(Y) M. If in addition Y « Yx X, then M.
PROOF. Let W be an element of U or Uk contained in Y. For any g G Nx(Y),
W9
lies in V, hence in M, and so g G M. Since g was arbitrary, Nx(Y) M, the
first assertion. This implies inductively that the terms of a subnormal chain from
Y to Y\ all lie in M, and the second assertion follows as well.
As a corollary, we obtain
LEMMA
1.5. If R is a p-subgroup of X which contains an element ofU orUk
for some k, then R M. Moreover if H X with H containing an element ofU
orUk for some k, then M contains a Sylow p-subgroup of H.
PROOF.
Set R0 = RDM. By definition elements of U and Uk lie in M, so B
0
contains such an element. Since i?o « B, the preceding lemma yields R M. For
the second statement, let R G Sylp(B) with R containing an element of U or Uk-
Then R M by the first statement, as required.
Taking H = X in the final statement of the preceding lemma, and using Lemma
1.4 as well, yields:
LEMMA
1.6. IfU ^ 0 then M contains a Sylow p-normalizer of X.
Using the Alperin-Goldschmidt theorem we also obtain the following result on
control of strong fusion. If P G Sylp(X) and P M X, we say that M controls
strong X-fusion in P if and only if for any subset A of P and element g G X such
that
A9
C P, there exists me M such that
a9
=
a171
for all a G A.
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