On <i>ss</i>-quasinormal and weakly <i>s</i>-supplemented subgroups of finite groups

Abstract

Suppose that $G$ is a finite group and $H$ is a subgroup of $G$. $H$ is called $ss$-quasinormal in $G$ if there is a subgroup $B$ of $G$ such that $G = HB$ and $H$ permutes with every Sylow subgroup of $B$; $H$ is called weakly $s$-supplemented in G if there is a subgroup T of G such that $G = HT$ and $H \bigcap T \leq H_{sG}$, where $H_{sG}$ is the subgroup of $H$ generated by all those subgroups of $H$ which are $s$-quasinormal in $G$. In this paper we investigate the influence of $ss$-quasinormal and weakly $s$-supplemented subgroups on the structure of finite groups. Some recent results are generalized and unified.