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Elementary equivalence for finitely generated nilpotent groups and multilinear maps

Published online by Cambridge University Press:  17 April 2009

Francis Oger
Francis Oger
Affiliation:
Equipe de Logique Mathematique, 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

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We show that two finitely generated finite-by-nilpotent groups are elementarily equivalent if and only if they satisfy the same sentences with two alternations of quantifiers. For each integer n ≥ 2, we prove the same result for the following classes of structures:

(1) the (n + 2)-tuples (A1, …, An+1, f), where A1, …, An+1 are disjoint finitely generated Abelian groups and f: A1 × … × An → An+1 is a n-linear map;

(2) the triples (A, B, f), where A, B are disjoint finitely generated Abelian groups and f: An → B is a n-linear map;

(3) the pairs (A, f), where A is a finitely generated Abelian group and f: An → A is a n-linear map.

In the proof, we use some properties of commutative rings associated to multilinear maps.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 58 , Issue 3 , December 1998 , pp. 479 - 493
Copyright
Copyright © Australian Mathematical Society 1998

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