Propermutable characterizations of finite soluble $PST$-groups and $PT$-groups

Let $H$ and $X$ be subgroups of a group $G$. We say that a subgroup $H$ is $X$-propermutable in $G$ provided that there is a subgroup $B$ of $G$ such that $G=N_G(H)B$ and $H$ $X$-permutes (in the sense of [1]) with all subgroups of $B$. In this paper we analyze the influence of $X$-propermutable subgroups on the structure of a finite group $G$. In particular, it is proved that $G$ is a soluble $PST$-group if and only if all Hall subgroups and all maximal subgroups of every Hall subgroup of $G$ are $X$-propermutable in $G$, where $X=Z_\infty(G)$.