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Journal of Group Theory

Editor-in-Chief: Parker, Christopher W.

Managing Editor: Wilson, John S. / Khukhro, Evgenii I. / Kramer, Linus

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Volume 20, Issue 6


Group varieties not closed under cellular covers and localizations

Daniel Herden / Montakarn Petapirak / José L. Rodríguez
Published Online: 2017-08-17 | DOI: https://doi.org/10.1515/jgth-2017-0021


A group homomorphism e:HG is a cellular cover of G if for every homomorphism φ:HG there is a unique homomorphism φ¯:HH such that φ¯e=φ. Group localizations are defined dually. The main purpose of this paper is to establish 20 varieties of groups which are not closed under taking cellular covers. This will use the existence of a special Burnside group of exponent p for a sufficiently large prime p as a key witness. This answers a question raised by Göbel in [12]. Moreover, by using a similar witness argument, we can prove the existence of 20 varieties not closed under localizations. Finally, the existence of 20 varieties of groups neither closed under cellular covers nor under localizations is presented as well.

Dedicated to the memory of Rüdiger Göbel


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About the article

Received: 2017-02-21

Published Online: 2017-08-17

Published in Print: 2017-11-01

Funding Source: Ministerio de Economía y Competitividad

Award identifier / Grant number: MTM2013-42178-P

Award identifier / Grant number: MTM2016-76453-C2-2-P

Funding Source: Consejería de Economía, Innovación, Ciencia y Empleo, Junta de Andalucía

Award identifier / Grant number: FQM-213

The third named author was supported by project MTM2013-42178-P, funded by the Spanish Ministry of Economy and Competitiveness, A EI/FEDER grant MTM2016-76453-C2-2-P, and by the Junta de Andalucía Grant FQM-213.

Citation Information: Journal of Group Theory, Volume 20, Issue 6, Pages 1073–1088, ISSN (Online) 1435-4446, ISSN (Print) 1433-5883, DOI: https://doi.org/10.1515/jgth-2017-0021.

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