on weakly ss-quasinormal and hypercyclically embedded properties of finite groups

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 $hcap t$ 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.