On the normal cores of certain subgroups of nilpotent groups

Let G be a group and H a subgroup of finite index in G. Then of course H contains a G-invariant subgroup C such that G/C is finite. In attempting to establish results of a similar nature, where “finite” is replaced by, for example, “finitely generated”, one notices immediately that a quite differently stated hypothesis is required. One reasonable approach would be to consider subgroups H which are “f.g. embedded” in G—indeed, the notion of a polycyclic embedding was utilised by P. Hall in [1].