Formations and Products of $\mathrm F(G)$-Subnormal Subgroups of Finite Solvable Groups

A subgroup $H$ of a finite group $G$ is said to be $\mathrm F(G)$-subnormal if it is subnormal in $H\mathrm F(G)$, where $\mathrm F(G)$ is the Fitting subgroup of $G$. In the paper, the problem of whether or not a formation $\mathfrak F$ contains products of $\mathrm F(G)$-subnormal $\mathfrak F$-subgroups of finite solvable groups is studied. In particular, solvable saturated formations $\mathfrak F$ with this property are described. Formation properties of groups having three solvable $\mathrm F(G)$-subnormal subgroups with pairwise coprime indices are studied. The supersolvability of any group $G$ having three supersolvable $\mathrm F(G)$-subnormal subgroups whose indices in $G$ are pairwise coprime is proved.