$\Omega$-foliated formations and Fitting classes of finite groups

A new functional approach to the study of classes of groups is proposed, resulting in description of all formations and Fitting classes of finite groups in the language of functions. The $\Omega$-foliated formations $\Omega F(f,\varphi)$ and $\Omega$-foliated Fitting classes $\Omega F(f,\varphi)$ with satellite $f$ and direction $\varphi$ are constructed. To each satellite $f$ there corresponds an infinite set of various directions $\varphi$. One direction leads to the previously considered $\Omega$-composite formations. In this way the $\Omega$-canonical and $\Omega$-free formations and Fitting classes are obtained. For a fixed direction $\varphi$ the structure of the minimal satellite $f$ is obtained.