On $\mathcal{F}$-S-Supplemented Subgroups

Abstract:

      Let $\mathcal{F}$ be a class of groups. A subgroup $H$ of a group $G$ is called $\mathcal{F}$-S-supplemented in $G$ if there exists a subgroup $K$ of $G$ such that $G=HK$ and $K/K\cap{H_G}\in\mathcal{F}$, where $H_G=\bigcap_{g\in G}H^g$ is the maximal normal subgroup of $G$ contained in $H$. In this paper, By using $\mathcal{F}$-S-supplemented subgroups, we give some new criteria for the solvability, nilpotency and supersolvability of finite groups. By these results, we may get a series of corollaries, which contain some known results.