On finitely generated subgroups of some soluble groups
Let \({\mathcal H}\) denote the class of groups in which the intersection of two finitely generated subgroups is finitely generated, and \({\mathcal K}\) the class of groups G in which each finitely generated subgroup is polycyclic or of finite index in G. Among the many interesting results in this paper are the following: A. If G is nilpotent-by-abelian and has max-n, then the following are equivalent: (i) \(G\in {\mathcal H}\) (ii) \(G\in {\mathcal K}\) (iii) G'' is polycyclic and G/G''\(\in {\mathcal H}\). B. There exists a finitely generated, nilpotent-by-abelian, \({\mathcal H}\)-group G which is not in \({\mathcal K}\) and has G'' non-polycyclic. C. For soluble groups of finite rank the properties \({\mathcal H}\) and \({\mathcal K}\) coincide. D. If G is finitely generated, nilpotent-by-abelian, then G is in \({\mathcal K}\) if and only if every section of G is in \({\mathcal H}\).
intersection of two finitely generated subgroups
scientific article; zbMATH DE number 3973156