Elementary Equivalence and Profinite Completions: A Characterization of Finitely Generated Abelian-by-Finite Groups
Francis Oger · Proceedings of the American Mathematical Society · 1988
In this paper, we show that any finitely generated abelian-by-finite group is an elementary submodel of its profinite completion. It follows that two finitely generated abelian-by-finite groups are elementarily equivalent if and only if they have the same finite images. We give an example of two finitely generated abelian-by-finite groups $G,H$ which satisfy these properties while $G \times {\bf {Z}}$ and $G \times {\bf {Z}}$ are not isomorphic. We also prove that a finitely generated nilpotent-by-finite group is elementarily equivalent to its profinite completion if and only if it is abelian-by-finite.