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. This implies that separable commensurated subgroups of finitely generated groups are virtually normal. A stream of applications to separable subgroups, polycyclic groups, residually finite groups, groups acting on trees, lattices in products of trees and just-infinite groups then flows from this main result.

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关于可比子群的残差和有限闭包

子组的残差闭包H一组的G是所有虚拟正规子群的交集G包含H. 我们证明如果G由有限多个陪集生成H而如果H是相称的，则剩余闭合H在G几乎是正常的。这意味着有限生成群的可分可通约子群实际上是正态的。然后从这个主要结果流出了对可分离子群、多环群、剩余有限群、作用于树的群、树乘积中的格和刚好无限群的应用流。