On the structure of some groups having finite contranormal subgroups

Following J. S. Rose, a subgroup $H$ of the group $G$ is said to be contranormal in $G$, if $G=H^{G}$. In a certain sense, contranormal subgroups are antipodes to subnormal subgroups. We study the structure of Abelian-by-nilpotent groups having a finite proper contranormal $p$-subgroup.