Groups in which some primary subgroups are weakly s-supplemented?

Abstract

A subgroup $H$ of a group $G$ is called weakly s-supplemented in $G$ if there is a subgroup $T$ such that $G=HT$ and $H\cap T\le H_{sG}$, where $H_{sG}$ is the subgroup of $H$ generated by all those subgroups of $H$ which are s-permutable in $G$. The influence of primary weakly s-supplemented subgroups on the structure of finite groups is investigated. An open question promoted by Skiba is studied and some known results are generalized.