Tao Zhao - School of Science, Shandong University of Technology

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.

‎s-permutable‎, ‎weakly SS-quasinormal‎, p-nilpotent‎, ‎hypercyclically embedded