Abstract:

For  saturated formations $\mathfrak{X}\subseteq \mathfrak{F}$
where  $\mathfrak{F}$ is hereditary it is proved that the class $\mathfrak{X}_{sn_\mathfrak{F}}$ of groups all of whose $\mathfrak{X}$-projectors are $\mathfrak{F}$-subnormal is a saturated formation and its  local definitions are found. The methods for calculating the $\mathfrak{X}_{sn_\mathfrak{F}}$-residual of a finite solvable group are discussed in the paper.