Finite Groups with Weakly $s$-Semipermutable Subgroups

Abstract

Suppose $G$ is a finite group and $H$ is a subgroup of $G$. $H$ is said to be $s$-semipermutable in $G$ if $H{G}_p=G_{p}H$ for any Sylow $p$-subgroup $G_p$ of $G$ with $(p,|H|)=1$; $H$ is called weakly $s$-semipermutable in $G$ if there is a subgroup $T$ of $G$ such that $G=HT$ and $H\cap T$ is $s$-semipermutable in $G$. We investigate the influence of weakly $s$-semipermutable subgroups on the structure of finite groups. Some recent results are generalized and unified.