Polycyclic groups, finite images, and elementary equivalence (Q914824)

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.