## Finite Groups with Weakly $s$-Permutably Embedded and Weakly $s$-Supplemented Subgroups

### Volume 59 / 2011

Bulletin Polish Acad. Sci. Math. 59 (2011), 41-52
MSC: Primary 20D10; Secondary 20D20.
DOI: 10.4064/ba59-1-6

#### Abstract

Suppose $G$ is a finite group and $H$ is a subgroup of $G$. $H$ is called weakly $s$-permutably embedded in $G$ if there are a subnormal subgroup $T$ of $G$ and an $s$-permutably embedded subgroup $H_{se}$ of $G$ contained in $H$ such that $G=HT$ and $H\cap T\leq H_{se}$; $H$ is called weakly $s$-supplemented in $G$ if there is a subgroup $T$ of $G$ such that $G=HT$ and $H\cap T\leq H_{sG}$, where $H_{sG}$ is the subgroup of $H$ generated by all those subgroups of $H$ which are $s$-permutable in $G$. We investigate the influence of the existence of $s$-permutably embedded and weakly $s$-supplemented subgroups on the structure of finite groups. Some recent results are generalized.