On weakly $SS$-quasinormal and hypercyclically embedded properties of finite groups

Document Type: Research Paper

Author

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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