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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
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 $ 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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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1988-0954980-0
PII: S 0002-9939(1988)0954980-0
Article copyright: © Copyright 1988 American Mathematical Society



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