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Schur's theorem and its relation to the closure properties of the non-abelian tensor product

Part of: Homological algebra

Published online by Cambridge University Press:  26 January 2019

G. Donadze ,
M. Ladra  and
P. Páez-Guillán
G. Donadze
Affiliation:
Vladimir Chavchanidze Institute of Cybernetics of the Georgian Technical University, Sandro Euli str. 5, Tbilisi 0186, Georgia (gdonad@gmail.com)
M. Ladra
Affiliation:
Department of Matemáticas, University of Santiago de Compostela, 15782 Santiago de Compostela, Spain (manuel.ladra@usc.es; pilar.paez@usc.es)
P. Páez-Guillán
Affiliation:
Department of Matemáticas, University of Santiago de Compostela, 15782 Santiago de Compostela, Spain (manuel.ladra@usc.es; pilar.paez@usc.es)
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Abstract

We show that the Schur multiplier of a Noetherian group need not be finitely generated. We prove that the non-abelian tensor product of a polycyclic (resp. polycyclic-by-finite) group and a Noetherian group is a polycyclic (resp. polycyclic-by-finite) group. We also prove new versions of Schur's theorem.

Keywords

Non-abelian tensor productNoetherian groupSchur multiplier

MSC classification

Primary: 18G10: Resolutions; derived functors
Secondary: 18G50: Nonabelian homological algebra
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 2 , April 2020 , pp. 993 - 1002
Copyright
Copyright © Royal Society of Edinburgh 2019

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