General Lemmas
This chapter is a brief preliminary to the main events, which are the theorems
of Chapters 2 and 3. While most of the general group-theoretic results underlying
this volume have been presented in [I^], a bit of additional terminology and theory
will be required, and we devote this chapter to a handful of topics appropriate to
the main theory to be developed in the next two chapters1. That theory requires
in addition a more substantial background of preliminary 3C-group lemmas, which
will be found in our fourth chapter2 [II4].
1. Uniqueness Elements
There are a number of different but related notions of "uniqueness elements",
of which the following are the most important for us.
DEFINITION 1.1. Let X be a group, M a proper subgroup of X and p a prime.
We define
C(X, M,p) = {xe 1P(M) | Cx(x) M};
F(X, M,p) = {x e XP(M) | M controls the X-fusion of x in some Sylow
p-subgroup of M containing x}\ and
U(X, M,p) = C(X, M,p) n T{X, M,p).
The context will usually make it unmistakeable what X, M and p are, and
then we shall write
(1A) C = C(X,M,p), T = T{X,M,p) trndU =U(X,M,p).
There is a single condition which gives an equivalent definition of U:
(IB) U = {xe IP(M) | for any g G X, x9 G M if and only if g e M}.
Indeed if x belongs to the right side of this equation, then it is clear that Cx (x) M
and that any X-conjugate of x in M is already an M-conjugate of x. Conversely
suppose that x G U, g G X and y = x9 £ M. Since x G T, the X-fusion of x
in some Sylow p-subgroup P of M is controlled in M. There exists h\ G M such
remind the reader that the preceding chapters will be referenced as follows: Overview
[Ii], Outline of Proof [I2], General Group Theory [IQ] and Almost Simple X-Groups [1^4]. The
first two of these comprise the first volume of this series, and the other two comprise volumes 2
and 3, respectively. The various chapters of the current volume belong to Part II and so will be
referenced as [Hi] (the current chapter), [II2], etc.
In the chain of logic, Chapter 4 lies between Chapters 1 and 2, but we have placed that
resource in what seems to us a more natural location for the reader.
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