University of IsfahanInternational Journal of Group Theory2251-76503420141201On weakly $SS$-quasinormal and hypercyclically embedded properties of finite groups1725495010.22108/ijgt.2014.4950ENTaoZhaoSchool of Science, Shandong University of TechnologyJournal Article20140216A subgroup $H$ is said to be $s$-permutable in a group $G$, if $HP=PH$ holds for every Sylow subgroup $P$ of $G$. If there exists a subgroup $B$ of $G$ such that $HB=G$ and $H$ permutes with every Sylow subgroup of $B$, then $H$ is said to be $SS$-quasinormal in $G$. In this paper, we say that $H$ is a weakly $SS$-quasinormal subgroup of $G$, if there is a normal subgroup $T$ of $G$ such that $HT$ is $s$-permutable and $H\cap T$ is $SS$-quasinormal in $G$. By assuming that some subgroups of $G$ with prime power order have the weakly $SS$-quasinormal properties, we get some new characterizations about the hypercyclically embedded subgroups of $G$. A series of known results in the literature are unified and generalized.https://ijgt.ui.ac.ir/article_4950_c0915a41877e3a4bb1db406fbaca42cf.pdf