The residual closure of a subgroup $H$ of a group $G$ is the intersection of all virtually normal subgroups of $G$ containing $H$. We show that if $G$ is generated by finitely many cosets of $H$ and if $H$ is commensurated, then the residual closure of $H$ in $G$ is virtually normal. Various applications to separable subgroups, polycyclic groups, residually finite groups, groups acting on trees, lattices in products of trees and just-infinite groups are described. 查看全文>>