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Licensed Unlicensed Requires Authentication Published by De Gruyter October 17, 2017

Uncountable locally free groups and their group rings

  • Tsunekazu Nishinaka EMAIL logo
From the journal Journal of Group Theory

Abstract

In this note, we show that an uncountable locally free group G, and therefore every locally free group, has a free subgroup whose cardinality is the same as that of G. This result directly improves the main result in [4] and establishes the primitivity of group rings of locally free groups.


Communicated by Pavel A. Zalesskii


Funding statement: This research was partially supported by Grants-in-Aid for Scientific Research (KAKEN) under grant no. 26400055.

Acknowledgements

I am grateful to the Editor and an anonymous reviewer for the helpful suggestions that improved the exposition of this paper.

References

[1] G. Higman, A finitely related group with an isomorphic proper factor group, J. Lond. Math. Soc. 26 (1951), 59–61. 10.1112/jlms/s1-26.1.59Search in Google Scholar

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[3] A. I. Mal’cev, On groups of finite rank, Mat. Sbornik N.S. 22(64) (1948), 351–352. Search in Google Scholar

[4] T. Nishinaka, Group rings of countable non-abelian locally free groups are primitive, Internat. J. Algebra Comput. 21 (2011), no. 3, 409–431. 10.1142/S0218196711006273Search in Google Scholar

[5] M. Takahasi, Note on locally free groups, J. Inst. Polytech. Osaka City Univ. Ser. A. Math. 1 (1950), 65–70. Search in Google Scholar

Received: 2016-1-2
Revised: 2017-8-18
Published Online: 2017-10-17
Published in Print: 2018-1-1

© 2017 Walter de Gruyter GmbH, Berlin/Boston

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