Polycyclic groups, finite images, and elementary equivalence
This paper settles the following question: If G is a polycyclic group and H is a finitely generated group which is elementarily equivalent to G, must H be polycyclic? An affirmative answer follows from the paper's main result, which is purely algebraic in character. It asserts that, if G is polycyclic and H is a finitely generated residually finite group having exactly the same finite images as G, then H is polycyclic. The proof starts with a study of certain abelian groups and reduces the general case to the metabelian case. \(\forall \exists\) sentences provide the link between logic and algebra.
finitely generated residually finite group
scientific article; zbMATH DE number 4150470