On the theory of fitting classes of finite soluble groups

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