Finite Factorizable Groups with $\mathbb P$-Subnormal $\mathrm v$-Supersolvable and $\mathrm{sh}$-Supersolvable Factors

Abstract

We study a finite factorized group $G=AB$ in the case when the factors $A$ and $B$ can be connected to $G$ by a chain of subgroups with prime indices, and either all subgroups with nilpotent derived subgroups or all Schmidt subgroups in $A$ and $B$ are supersolvable. Such factorizations cover both the groups that are products of normal supersolvable subgroups and mutually permutable products of supersolvable subgroups. In particular, it follows from the results obtained here that all Schmidt subgroups in products of normal supersolvable subgroups and in mutually permutable products of supersolvable subgroups are supersolvable; however, a nonsupersolvable subgroup with nilpotent derived subgroup can exist.