Finite groups with some weakly $s$-permutably embedded and weakly $s$-supplemented subgroups

Abstract

Let $G$ be a finite group. A subgroup $H$ of $G$ is called weakly $s$-permutably embedded in $G$ if there is a subnormal subgroup $T$ of $G$ and an $s$-permutably embedded subgroup $H_{se}$ of $G$ contained in $H$ such that $G=HT$ and $H\cap T\le H_{se}$. The subgroup $H$ is called weakly $s$-supplemented in $G$ if $G$ has a subgroup $K$ such that $HK=G$ and $H\cap K\le H_{sG}$, where $H_{sG}$ is the largest $s$-permutable subgroup of $G$ contained in $H$. In this paper, we investigate the influence of weakly $s$-permutably embedded and weakly $s$-supplemented subgroups on the structure of finite groups. Some recent results are generalized.