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

Author:
Francis Oger

Journal:
Proc. Amer. Math. Soc. **103** (1988), 1041-1048

MSC:
Primary 03C60; Secondary 20A10, 20F18, 20F19

DOI:
https://doi.org/10.1090/S0002-9939-1988-0954980-0

MathSciNet review:
954980

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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 which satisfy these properties while and 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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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1988-0954980-0

Article copyright:
© Copyright 1988
American Mathematical Society