Constructing regular saturated formations of finite soluble groups

For a formation $\mathfrak{F}$ of finite groups consider a graph whose vertices are elements of a finite group and two vertices are connected by an edge if and only if they generates non-$\mathfrak{F}$-group as elements of a group. A hereditary formation $\mathfrak{F}$ is called regular if the set of all isolated vertices of the described graph coincides with the intersection of all maximal $\mathfrak{F}$-subgroups in every group. The constructive description of saturated regular formations of soluble groups which improves the results of Lucchini and Nemmi is obtained in this paper. In particular, it is showed that saturated regular non-empty formations of soluble groups are just hereditary formations $\mathfrak{F}$ of soluble groups that contains every group all whose cyclic primary subgroups are $K$-$\mathfrak{F}$-subnormal. Also we prove that the lattice of saturated regular formations of soluble groups is lattice isomorphic to the Steinitz’s lattice.