#### Vol. 8, No. 2, 2015

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

### Guo Zhong, XuanLong Ma, Shixun Lin, Jiayi Xia and Jianxing Jin

Vol. 8 (2015), No. 2, 253–262
##### 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.

##### Keywords
weakly $s$-permutably embedded subgroups, weakly $s$-supplemented subgroups, $p$-nilpotent groups
Primary: 20D10
Secondary: 20D20