$\omega$-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 $\varphi$ are introduced. A direction $\varphi$ is defined as a mapping of the set $\mathbb P$ 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 $\omega$-fibered formation with a direction $\varphi$ such that $\varphi (p)=\mathfrak G_{p'}\mathfrak N_p$ for any prime $p$. In the paper we study some basic properties of $\omega$-fibered formations and of $\omega$-fibered Fitting classes with direction $\varphi$ and obtain the structure of their minimal satellites for a given $\varphi$.