Weakly $S$-semipermutable subgroups and $p$-nilpotency of groups

Abstract

A subgroup $H$ of a finite group $G$ is said to be $S$-semipermutable in $G$ if $H{G}_p=G_{p}H$ for every Sylow subgroup $G_p$ of $G$ with $(|H|,p)=1$. A subgroup $H$ of $G$ is said to be weakly $S$-semipermutable in $G$ if there exists a normal subgroup $T$ of $G$ such that $HT$ is $S$-permutable and $H\cap T$ is $S$-semipermutable in $G$. In this paper we prove that for a finite group $G$, if some cyclic subgroups or maximal subgroups of $G$ are weakly $S$-semipermutable in $G$, then $G$ is $p$-nilpotent.