#### 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
##### Milestones
Revised: 6 August 2013
Accepted: 7 August 2013
Published: 3 March 2015

Communicated by Joseph A. Gallian
##### Authors
 Guo Zhong School of Mathematical Sciences Guangxi Teachers Education University Nanning, Guangxi 530023 China XuanLong Ma School of Mathematical Sciences Beijing Normal University Beijing 100875 China Shixun Lin College of Mathematics and Statistics Zhaotong University Zhaotong, Yunnan 657000 China Jiayi Xia School of Mathematical Sciences Guangxi Teachers Education University Nanning, Guangxi 530023 China Jianxing Jin School of Mathematical Sciences Guangxi Teachers Education University Nanning, Guangxi 530023 China