On SS-Quasinormal and S-Quasinormally Embedded Subgroups of Finite Groups

A subgroup $H$ of a group $G$ is said to be an SS-quasinormal (Supplement-Sylow-quasinormal) subgroup if there is a subgroup $B$ of $G$ such that $HB = G$ and $H$ permutes with every Sylow subgroup of $B$. A subgroup $H$ of a group $G$ is said to be S-quasinormally embedded in $G$ if for every Sylow subgroup $P$ of $H$, there is an S-quasinormal subgroup $K$ in $G$ such that $P$ is also a Sylow subgroup of $K$. Groups with certain SS-quasinormal or S-quasinormally embedded subgroups of prime power order are studied.