^{}School of Science, Shandong University of Technology

Abstract

A 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.

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