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Elementary equivalence for abelian-by-finite and nilpotent groups

Published online by Cambridge University Press:  12 March 2014

Francis Oger
Francis Oger*
Affiliation:
Équipe de Logique Mathématique, Université Paris, VII — C.N.R.S., 2 Place Jussieu — Case 7012, 75251 Paris Cédex 05, France, E-mail: oger@logique.jussieu.fr
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Abstract

We show that two abelian-by-finite groups are elementarily equivalent if and only if they satisfy the same sentences with two alternations of quantifiers. We also prove that abelian-by-finite groups satisfy a quantifier elimination property. On the other hand, for each integer n, we give some examples of nilpotent groups which satisfy the same sentences with n alternations of quantifiers and do not satisfy the same sentences with n + 1 alternations of quantifiers.

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 66 , Issue 3 , September 2001 , pp. 1471 - 1480
Copyright
Copyright © Association for Symbolic Logic 2001

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References

REFERENCES

[1]Belegradek, O. V., The model theory of unitriangular groups, Annals of Pure and Applied Logic, vol. 68 (1994), pp. 225–271.CrossRefGoogle Scholar
[2]Burris, S., Bounded boolean powers and ≡n, Algebra Universalis, vol. 8 (1978), pp. 137–138.CrossRefGoogle Scholar
[3]Felgner, U., The model theory of FC-groups, Mathematical logic in Latin America, Studies in Logic and the Foundations of Mathematics, no. 99, North-Holland, Amsterdam, 1980.Google Scholar
[4]Kaye, R., Diophantine induction, Annals of Pure and Applied Logic, vol. 46 (1990), pp. 1–40.CrossRefGoogle Scholar
[5]Kaye, R., Models ofPeano arithmetic, Oxford logic guides, no. 15, Clarendon Press, Oxford, 1991.Google Scholar
[6]Mal'cev, A. I., The metamathematics of algebraic systems, Studies in logic and the foundations of mathematics, no. 66, North-Holland, Amsterdam, 1971.Google Scholar
[7]Oger, F., Elementary equivalence and profinite completions: A characterization of finitely generated abelian-by-finite groups, Proceedings of the American Mathematical Society, vol. 103 (1988), pp. 1041–1048.CrossRefGoogle Scholar
[8]Oger, F., Axiomatization of abelian-by-G groups for a finite group G, Archive for Mathematical Logic, to appear.Google Scholar
[9]Oger, F., Elementary equivalence for finitely generated nilpotent groups and multilinear maps, Bulletin of the Australian Mathematical Society, vol. 58 (1998), pp. 479–493.CrossRefGoogle Scholar
[10]Prest, M., Model theory and modules, London mathematical society lecture notes series, no. 130, Cambridge University Press, Cambridge, 1988.Google Scholar
[11]Robinson, D. J. S., A course in the theory of groups, Graduate texts in mathematics, no. 80, Springer, Berlin, 1982.Google Scholar
[12]Szmielew, W., Elementary properties of abelian groups, Fundamenta Mathematicae, vol. 41 (1955), pp. 203–271.CrossRefGoogle Scholar
[13]Waszkiewicz, J., ∀n-theories of boolean algebras, Colloquium Mathematicum, vol. 30 (1974), pp. 171–175.CrossRefGoogle Scholar