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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Tarski's problem for solvable groups


Authors: Pat Rogers, Howard Smith and Donald Solitar
Journal: Proc. Amer. Math. Soc. 96 (1986), 668-672
MSC: Primary 03C60; Secondary 20A15, 20E05
DOI: https://doi.org/10.1090/S0002-9939-1986-0826500-0
MathSciNet review: 826500
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Abstract: In this paper, we show that the free solvable groups (as well as the free nilpotent groups) of finite rank have different elementary theories (i.e., they do not satisfy the same first order sentences of group theory). This result is obtained using a result in group theory (probably due to Malcev and following immediately from a theorem of Auslander and Lyndon) that, for a free nontrivial solvable group, the last nontrivial group in its derived series is is own centralizer.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1986-0826500-0
Keywords: Free nilpotent group, free solvable group, elementarily equivalent, basic commutator
Article copyright: © Copyright 1986 American Mathematical Society