4
PART I, CHAPTER G: GENERAL GROUP THEORY
acts by automorphisms on a group F, we implicitly form the semidirect product
YX and therefore can use such symbols as Cx (Y) and [Y, X].
As described in [l2;§37], we introduce certain abbreviated notation for local
subgroups of the ambient group G. Thus for any subgroup or subset Y of G, we
write Cy for
CQ{Y)
and Ny for
NG(Y).
We do this only for G itself, and make
no abbreviation for Nx(Y) or Cx(Y) for other groups X.
We shall have frequent occasion to consider groups of the form N = Nx (Y)
and N = Nx{Y)/Cx{Y) for some subgroup Y of X. Conjugation by elements of
N leads to an embedding of N in the automorphism group Aut(Y); the image of
this embedding is called AutxOO, and it will often be convenient to identify N
with Autx(Y), so that we may write N Aut(Y). Similarly, the image of N in
the outer automorphism group Out(Y) = Aut(Y')/Inn(Y') is called Outx(Y)- An
element x is said to invert the element y (or the subgroup Y) if and only if yx y~l
(or yx y~x for all y G Y). Here as always yx = x~lyx.
At several places we make use of graphs. By this term we always mean undi-
rected graphs. The greatest common divisor of integers m and n is written (ra,n).
The largest power of p dividing n is np. The finite field of cardinality q is ¥q.
Finally we mention that in later chapters (but not this one), we occasionally
use some elementary number-theoretic results, such as quadratic reciprocity and
Fermat's characterization of primes which are sums of two squares. We state also
the following well-known theorem of Zsigmondy ([GLl;32-9] or [Zl]):
THEOREM
1.1
(ZSIGMONDY).
If p is a prime and n 1, then
pn
- 1 has a
prime divisor not dividing pl 1 for any 1 i n, unless either p is a Mersenne
prime and n 2 or p 2 and n = 6.
2. Some DC-Group Conditions
For the most part, the results in this volume are proved just from the axioms
of group theory. However, for a few specific but crucial results, such as Lv -balance
and the nonsolvable case of the signalizer functor theorem, there are no known
proofs independent of an extra hypothesis that the simple sections of the groups
in question satisfy certain properties. These properties have in fact been verified
for UC-groups and so they have a curious status. On the one hand, they hold in
proper sections of a minimal counterexample to the Classification Theorem, so that
results that depend on them, like Lv -balance and the nonsolvable signalizer functor
theorem, are available in the proof of the classification. On the other hand, at the
end of the proof of the Classification Theorem they as well as their consequences
such as Lp/-balance will be corollaries and will have been proved for all finite
groups.
The irksome point is that these properties have as yet not been given a proof of
the most satisfactory kind—an a priori development independent of the full proof
of the classification. The only exception to this is Glauberman's proof of property
(G2) [G13]. We must settle for the proofs which we have, a sharp reminder indeed
of how much is yet to be learned about finite simple groups.
As this chapter is intended to be devoted not to DC-groups but to general phe-
nomena which can be proved without case-by-case verification, we shall formulate
the properties in this chapter but postpone their verification until the succeeding
chapter on almost simple DC-groups. In this chapter, whenever a theorem requires
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