Geodesic
On the normal cores of certain subgroups of nilpotent groups
Glasgow mathematical journal, Tome 36 (1994) no. 1, pp. 113-115

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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]. 
Smith, Howard. On the normal cores of certain subgroups of nilpotent groups. Glasgow mathematical journal, Tome 36 (1994) no. 1, pp. 113-115. doi: 10.1017/S0017089500030615
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[1] 1.Hall, P., Finiteness conditions for soluble groups, Proc. London Math. Soc. (3) 4 (1954), 419–436. Google Scholar

[2] 2.Hall, P., The Edmonton notes on nilpotent groups (QMC Mathematics Notes 1969). Google Scholar

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