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# Journal of Group Theory

Editor-in-Chief: Parker, Christopher W. / Wilson, John S.

Managing Editor: Khukhro, Evgenii I. / Kramer, Linus

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# 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

## Abstract

A group homomorphism $e:H\to G$ is a cellular cover of G if for every homomorphism $\phi :H\to G$ there is a unique homomorphism $\overline{\phi }:H\to H$ such that $\overline{\phi }e=\phi$. Group localizations are defined dually. The main purpose of this paper is to establish ${2}^{{\mathrm{\aleph }}_{0}}$ 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 ${2}^{{\mathrm{\aleph }}_{0}}$ varieties not closed under localizations. Finally, the existence of ${2}^{{\mathrm{\aleph }}_{0}}$ 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

Funding Source: Ministerio de Economía y Competitividad

Award identifier / Grant number: MTM2013-42178-P

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

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, ISSN (Online) 1435-4446, ISSN (Print) 1433-5883,

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© 2017 Walter de Gruyter GmbH, Berlin/Boston.

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