Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-25T02:02:21.171Z Has data issue: false hasContentIssue false
  • English
  • Français

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 ([email protected])
M. Ladra
Affiliation:
Department of Matemáticas, University of Santiago de Compostela, 15782 Santiago de Compostela, Spain ([email protected]; [email protected])
P. Páez-Guillán
Affiliation:
Department of Matemáticas, University of Santiago de Compostela, 15782 Santiago de Compostela, Spain ([email protected]; [email protected])
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Antony, A.E., Donadze, G., Sivaprasad, V.P. and Thomas, V.Z.. The second stable homotopy group of the Eilenberg Maclane space. Math. Z. 287 (2017), 1327–1342.CrossRefGoogle Scholar
2Brown, R. and Loday, J.-L.. Excision homotopique en basse dimension. C. R. Acad. Sci. Paris Sér. I Math. 298 (1984), 353–356.Google Scholar
3Brown, R. and Loday, J.-L.. Van Kampen theorems for diagrams of spaces. Topology 26 (1987), 311–335, with an appendix by M. Zisman.CrossRefGoogle Scholar
4Brown, R., Johnson, D.L. and Robertson, E.F.. Some computations of non-abelian tensor products of groups. J. Algebra 111 (1987), 177–202.CrossRefGoogle Scholar
5Dixon, M.R., Kurdachenko, L.A. and Pypka, A.A.. The theorems of Schur and Baer: a survey. Int. J. Group Theory 4 (2015), 21–32.Google Scholar
6Donadze, G., Ladra, M. and Thomas, V.. On the closure properties of the non-abelian tensor product. J. Algebra 472 (2017), 399–413.CrossRefGoogle Scholar
7Ellis, G.J.. The non-abelian tensor product of finite groups is finite. J. Algebra 111 (1987), 203–205.CrossRefGoogle Scholar
8Guin, D.. Cohomologie et homologie non abéliennes des groupes. J. Pure Appl. Algebra 50 (1988), 109–137.CrossRefGoogle Scholar
9Inassaridze, H.. Non-abelian homological algebra and its applications, Mathematics and its Applications 421 (Dordrecht: Kluwer Academic Publishers, 1997).CrossRefGoogle Scholar
10Moravec, P.. The non-abelian tensor product of polycyclic groups is polycyclic. J. Group Theory 10 (2007), 795–798.CrossRefGoogle Scholar
11Yu. Ol'shanskiĭ, A.. Geometry of defining relations in groups, Mathematics and its Applications 70 (Dordrecht: Kluwer Academic Publishers, 1991).CrossRefGoogle Scholar
12Robinson, D.. A course in the theory of groups, 2nd edn, GTM 80 (New York: Springer, 1996).CrossRefGoogle Scholar
13Rotman, J.J.. An introduction to the theory of groups, 4th edn, GTM 148 (New York: Springer, 1995).CrossRefGoogle Scholar
14Visscher, M.P.. On the nilpotency class and solvability length of nonabelian tensor products of groups. Arch. Math. (Basel) 73 (1999), 161–171.CrossRefGoogle Scholar