Abstract
We consider a finiteness condition on centralizers in a group G, namely that \(|C_G(x):\langle x \rangle |<\infty \) for every \(\langle x \rangle \ntriangleleft G\). For periodic groups, this is the same as \(|C_G(x)|<\infty \) for every \(\langle x \rangle \ntriangleleft G\). We give a full description of locally finite groups satisfying this condition. As it turns out, they are a special type of cyclic extensions of Dedekind groups. We also study a variation of our condition, where the requirement of finiteness is replaced with a bound: \(|C_G(x):\langle x \rangle |\le n\) for every \(\langle x \rangle \ntriangleleft G\), for some fixed n. In this case, we are able to extend our analysis to the class of periodic locally graded groups.
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Acknowledgments
We wish to thank Professor D. J. S. Robinson for interesting discussions and helpful comments.
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Communicated by A. Constantin.
Gustavo A. Fernández-Alcober and Leire Legarreta are supported by the Spanish Government, Grants MTM2011-28229-C02-02 and MTM2014-53810-C2-2-P, and by the Basque Government, Grants IT753-13 and IT974-16. Antonio Tortora and Maria Tota would like to thank the Department of Mathematics at the University of the Basque Country for its excellent hospitality while part of this paper was being written; they also wish to thank G.N.S.A.G.A. (INdAM) for financial support.
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Fernández-Alcober, G.A., Legarreta, L., Tortora, A. et al. A finiteness condition on centralizers in locally finite groups. Monatsh Math 183, 241–250 (2017). https://doi.org/10.1007/s00605-016-0939-4
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DOI: https://doi.org/10.1007/s00605-016-0939-4