The influence of partially $tau$-quasinormal subgroups on the structure of finite groups

A subgroup $H$ of a group $G$ is said to be $\tau$-quasinormal in $G$ if $H$ permutes with every Sylow subgroup $Q$ of $G$ such that $insert ignore into journalissuearticles values(|H|,|Q|);=1$ and $insert ignore into journalissuearticles values(|H|,|Q|^G);\neq1$; $H$ is called partially $\tau$-quasinormal in $G$ if $G$ has a normal subgroup $T$ such that $HT$ is $S$-quasinormal in $G$ and $H\cap T\leq H_{\tau G}$, where $H_{\tau G}$ is the subgroup generated by all those subgroups of $H$ which are $\tau$-quasinormal in $G$. In this paper, we investigate the influence of some partially $\tau$-quasinormal subgroups on the structure of finite group. Some new characterizations of $p$-supersoluble and $p$-nilpotent groups are obtained.