Abstract
We prove that a residually finite group G satisfying an identity \(w\equiv 1\) and generated by a commutator closed set X of bounded left Engel elements is locally nilpotent. We also extend such a result to locally graded groups, provided that X is a normal set. As an immediate consequence, we obtain that a locally graded group satisfying an identity, all of whose elements are bounded left Engel, is locally nilpotent.
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Acknowledgements
The authors wish to thank Professor P. Shumyatsky for interesting discussions and the anonymous referee for useful comments.
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R. Bastos was partially supported by FAPDF/Brazil. N. Mansuroǧlu was supported by the Scientific and Technological Research Council of Turkey (TUBITAK) under the programme TUBITAK 2219-International Postdoctoral Research Fellowship and she would like to thank the Department of Mathematics at the University of Salerno for its excellent hospitality. A. Tortora and M. Tota are members of G.N.S.A.G.A. (INdAM).
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Bastos, R., Mansuroğlu, N., Tortora, A. et al. Bounded Engel elements in groups satisfying an identity. Arch. Math. 110, 311–318 (2018). https://doi.org/10.1007/s00013-017-1137-x
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DOI: https://doi.org/10.1007/s00013-017-1137-x