ω-Fibered formations and Fitting classes of finite groups

In this paper, we suggest a new functional approach to the study of group classes which enables us to describe all formations and Fitting classes of finite groups in the language of functions. The notions of omega-fibered formation and of omega-fibered Fitting class with direction phi are introduced. A direction phi is defined as a mapping of the set P of all primes into the set of all nonempty Fitting formations. The existence of infinitely many mappings of this kind makes it possible to construct new forms of formations and Fitting classes for a given nonempty set omega. In particular, an omega-local formation is an w-fibered formation with a direction p such that phi(p) = B-p'N-p for any prime p. In the paper we study some basic properties of omega-fibered formations and of omega-fibered Fitting classes with direction p and obtain the structure of their minimal satellites for a given phi.