Elementary equivalence and profinite completions: a characterization of finitely generated abelian-by-finite groups

Oger, Francis

doi:10.1090/S0002-9939-1988-0954980-0

Elementary equivalence and profinite completions: a characterization of finitely generated abelian-by-finite groups
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Proc. Amer. Math. Soc. 103 (1988), 1041-1048 Request permission

Abstract:

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.

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