Classes of Finite Soluble Groups
I want to give here a rather biased account of recent work in the theory of classes of finite soluble groups. I will be concentrating on results which have something to say about the classes themselves, rather than results which use the classes to obtain
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CANBERRA 1973, pp. 226-237.
CLASSES OF FINITE SOLUBLE GROUPS
1.
Closed classes
I want to give here a rather biased aeeount of reeent work in the theory of elasses of finite soluble groups.
I will be eoneentrating on results whieh have
something to say about the elasses themselves, rather than results whieh use the elasses to obtain a pieture of the internal strueture of finite soluble groups.
My
main exeuse for doing so is that this part of the theory is at a very interesting stage:
the elasses are proving to be more exotie than might have been expeeted, and
though we know little about them, some results and teehniques are appearing, and it seems likely we will not remain so ignorant for long. The theory started with saturated formations in 1963, though a number of earlier papers gave hints of what was coming: [1], and Carter [12].
see for example the papers of Hall [29], Baer
Gaschütz introduced saturated formations in [24] to give a
unified aceount of the theories of Hall [28] and Carter, [13] and [14].
Hall and
Carter established the existence of certain characteristie conjugacy elasses of subgroups in finite soluble groups (the Hall and Carter subgroups). We start with formations:
a formation
~
is a class of groups (from now on,
all groups will be finite and soluble) such that
and
Q!~!
RO!~!'
Q!
=
{G
G is an epimorphie image of some H E ~
Ra!
=
{G
G ~ Ni' i
=
1, "', n, GINi E!, nNi
=
where
I}
A formation is said to be saturated if in addition Eq)1 = {G : GI> N, N ~ ~G, GIN E ~
Two easy examp1es of saturated formations are the class some set
11
S
=Tl
of primes, and the e1ass of all nilpotent groups
of all
1I-groups, for
!.
The fundamental
theorem of Gaschütz and Lubeseder ([27] and [41]) that a formation is saturated if and only if it has a loeal definition is the most important single tool for proving results about and eonstrueting examples of saturated formations:
for an aceount of
M. F. Newman (ed.), Proceedings of the Second International Conference on the Theory of Groups © Springer-Verlag Berlin Heidelberg 1974
Classes of finite sol uble groups
227
this theorem, see Huppert [35] VI, 7. It was the following property that gave saturated formations their importance.
G be a group, and
Let
of
~-projector
of
G,
if
G
a class of groups:
~
is a maximal
H
is a maximal
NH/N
is a saturated formation, With
~
=~
G an
and for every normal subgroup
N
Gaschütz showed in [24] that if
~
always exist, and they are all conjugate.
~-projectors
we get the Hall
G/N.
of
~-subgroup
H of
we call a subgroup
~-subgroup,
n-subgroups, and with
~
=!
we get the Carter
subgroups. At this stage, a natural question arises.
Call a class
for which
~
always exist and are all conjugate a projective class:
~-projectors
projective class is it a saturated formation?
negatively (45), and gave a characterisation of projective classes: called Schunck classes.
1
Denote by
if
~
is a
Schunck answered this question they are now
the class of all pri
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