Lockett, Francis Peter (1971) On the theory of fitting classes of finite soluble groups. PhD thesis, University of Warwick.

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Abstract

We continue the study Fitting classes begun by Fischer in
1966 and carried on by (notably) Gaschütz and Hartley.
Disappointingly the theory has, as yet, failed to display the
richness of its predecessor, the theory of Formations. Here we
present our contributions, embedded in a survey of the progress
so far made in this tantalizing part of finite soluble group
theory.
Chapter 0 indicates the group theoretic notation we use,
while Chapter 1 contains the basic results and terminology of
Fitting class theory. Broadly speaking this theory comprises a
study of the classes themselves, a study of the embedding of the
F-subgroups (subgroups which belong to F) of an arbitrary
group G and a study of the relation between F and the F-subgroups
of G. As in Formation theory we focus attention on
canonical sets of
F-subgroups, namely the F-injectors, the
Fischer F-subgroups and the maximal F-subgroups containing the
radical.
Chapter 2 begins with analyses of several examples of Fitting
classes, establishing the coincidence of these three sets of
F-subgroups (in all groups) in many cases, a property not enjoyed
by all Fitting classes. Here too we examine some known 'new
classes from old' procedures, and introduce a new one Fπ
(defined for any
F
and set of primes π), showing how this concept
may be used to characterize the injectors for the product
F1F2
of two Fitting classes. The chapter ends with some remarks on
the thorny problem of generating Fitting classes from given groups
and we present an imitation of work of Dark, the only person to
achieve progress in this direction. Finally we show how one of
the classes so constructed settles a question posed by Gaschütz.
Chapter 3 develops the theory of pronormal subgroups, based
on key theorems of Mann-Alperin and Fischer, showing in
particular that a permutable product of pronormal subgroups is
again pronormal. This approach yields a more compact version of
work on subgroups of a group which are p-normally embedded for
all primes p (we use the term strongly pronormal), published by
Chambers. The injectors for a Fischer class (in particular a
subgroup closed Fitting class) have this property.
Chapter 4 attacks the problem of determining the injectors
for the class Fπ and shows, in the light of chapter 3, that
the natural guess (a product of an F-injector and a Hall π'-
subgroup) holds good when, for instance,
F
is a Fischer class.
However, modification of the example of Dark denies that this is
in general the case. So arises the concept of permutability of
a Fitting class and, after giving a new proof of a related lemma
of Fischer, we establish conditions on a Fitting class equivalent
to its being permutable, involving system normalizers.
Chapter 5 takes a preliminary look at the analogue of Cline's
partial ordering of strong containment («) for Fitting classes,
and we show that a Fitting class maximal in this sense and having
strongly pronormal injectors in all groups, is necessarily a
normal Fitting class. In our final section we examine the
radical of a direct power of a group G and show that, for a normal
class, the radical is never the corresponding direct power of the
radical of G (unless of course G lies in the class). This
investigation puts the set of normal Fitting classes in a new
setting, and we demonstrate that to each Fitting class
F there
corresponds a well defined class F* with properties close to
those of F.

Item Type:	Thesis (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Finite groups
Official Date:	1971
Dates:	Date Event 1971 Submitted
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Hawkes, Trevor O., 1936-
Sponsors:	Science Research Council (Great Britain) (SRC) ; University of Warwick
Extent:	91 p.
Language:	eng

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