Finite widely $c$-supersoluble groups and their mutually permutable products

A group is $c$-supersoluble if all its chief factors are isomorphic to simple groups. We introduce the notion of widely $c$-supersoluble group, which generalizes $c$-supersolubility. We obtain the properties and find applications of products of mutually permutable subgroups. In particular, we establish the wide $c$-supersolubility of a finite group that is a product of two mutually permutable $c$-supersoluble subgroups of coprime indices.