Abstract
We give a detailed description of infinite locally nilpotent groups G such that the index \(|C_G(x):\langle x\rangle |\) is finite, for every \(\langle x \rangle \ntriangleleft G\). We are also able to extend our analysis to all non-periodic groups satisfying a variation of our condition, where the requirement of finiteness is replaced with a bound.
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Fernández-Alcober, G.A., Legarreta, L., Tortora, A., Tota, M.: A restriction on centralizers in finite groups. J. Algebra 400, 33–42 (2014)
Fernández-Alcober, G.A., Legarreta, L., Tortora, A., Tota, M.: Some restrictions on normalizers or centralizers in finite \(p\)-groups. Isr. J. Math. 208(1), 193–217 (2015)
Fernández-Alcober, G.A., Legarreta, L., Tortora, A., Tota, M.: A finiteness condition on centralizers in locally finite groups (2015, under review). arXiv:1510.03733 [math.GR]
Gromov, M.: Hyperbolic groups. In: Essays in Group Theory, pp. 75–263. Mathematical Sciences Research Institute Publications, vol. 8. Springer, New York (1987)
Lennox, J.C., Robinson, D.J.S.: The Theory of Infinite Soluble Groups. Oxford University Press, Oxford (2004)
Robinson, D.J.S.: Groups in which normality is a transitive relation. Proc. Camb. Philos. Soc. 60, 21–38 (1964)
Robinson, D.J.S.: Finiteness Conditions and Generalized Soluble Groups, Part 1. Springer, Berlin (1972)
Robinson, D.J.S.: A Course in the Theory of Groups, 2nd edn. Springer, New York (1996)
Shumyatsky, P.: Centralizers in locally finite groups. Turk. J. Math. 31, 149–170 (2007)
Shunkov, V.P.: On periodic groups with an almost regular involution. Algebra Log. 11, 260–272 (1972)
Zhang, Q., Gao, J.: Normalizers of nonnormal subgroups of finite \(p\)-groups. J. Korean Math. Soc. 49, 201–221 (2012)
Zhang, X., Guo, X.: Finite \(p\)-groups whose non-normal cyclic subgroups have small index in their normalizers. J. Group Theory 15, 641–659 (2012)
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Communicated by J. S. Wilson.
G. A. Fernández-Alcober and L. Legarreta are supported by the Spanish Government, Grants MTM2011-28229-C02-02 and MTM2014-53810-C2-2-P, and by the Basque Government, Grant IT753-13. A. Tortora and M. 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 nilpotent groups. Monatsh Math 182, 289–298 (2017). https://doi.org/10.1007/s00605-015-0854-0
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DOI: https://doi.org/10.1007/s00605-015-0854-0