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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A. Abdollahi, Engel elements in groups, In: Groups St Andrews 2009 in Bath. Volume 1, 94–117, Cambridge University Press, Cambridge, 2011.
L. Bartholdi, Algorithmic Decidability of Engel’s Property for Automaton Groups, Lecture Notes in Comput. Sci., 9691, Springer, Cham, 2016.
R. Bastos, On residually finite groups with Engel-like conditions, Comm. Algebra 44 (2016), 4177–4184.
R. Bastos, P. Shumyatsky, A. Tortora, and M. Tota, On groups admitting a word whose values are Engel, Int. J. Algebra Comput. 23 (2013), 81–89.
Y. de Cornulier and A. Mann, Some residually finite groups satisfying laws, In: Geometric Group Theory, 45–50, Trends Math., Birkhäuser, Basel, 2007.
J. D. Dixon, M. P. F. du Sautoy, A. Mann, and D. Segal, Analytic pro-p-groups, Cambridge University Press, Cambridge, 1991.
P. Hall and G. Higman, On the \(p\)-length of \(p\)-soluble groups and reduction theorems for Burnside’s problem, Proc. London Math. Soc. (3) 6 (1956), 1–42.
M. Lazard, Groupes analytiques \(p\)-adiques, IHES Publ. Math. 26 (1965), 389–603.
A. Lubotzky and A. Mann, Powerful \(p\)-groups. II. \(p\)-adic analytic groups, J. Algebra 105 (1987), 506–515.
L. Ribes and P. Zalesskii, Profinite Groups, 2nd edition, Springer-Verlag, Berlin, 2010.
E. Rips, Free Engel groups and similar groups, videos of the conference “Asymptotic properties of groups”, March 24–28, 2014; https://sites.google.com/site/geowalks2014/home/workshop.
D. J. S. Robinson, A Course in the Theory of Groups, 2nd edition, Springer-Verlag, New York, 1996.
P. Shumyatsky, Applications of Lie ring methods to group theory, In: Nonassociative Algebra and its Applications, R. Costa, A. Grishkov, H. Guzzo Jr., and L. A. Peresi (Eds.), 373–395, Lecture Notes in Pure and Appl. Math., Vol. 211, Dekker, New York, 2000.
P. Shumyatsky, Elements of prime power order in residually finite groups, Internat. J. Algebra Comput. 15 (2005), 571–576.
P. Shumyatsky, A. Tortora, and M. Tota, An Engel condition for orderable groups, Bull. Braz. Math. Soc. (N.S.) 46 (2015), 461–468.
P. Shumyatsky, A. Tortora, and M. Tota, On locally graded groups with a word whose values are Engel, Proc. Edinburgh Math. Soc. 59 (2016), 533–539.
P. Shumyatsky, A. Tortora, and M. Tota, On varieties of groups satisfying an Engel type identity, J. Algebra 447 (2016), 479–489.
J. Tits, Free subgroups in linear groups, J. Algebra 20 (1972), 250–270.
G. Traustason, Engel groups, In: Groups St Andrews 2009 in Bath Volume 2, 520–550, Cambridge University Press, Cambridge, 2011.
J. S. Wilson, Two-generator conditions for residually finite groups, Bull. London Math. Soc. 23 (1991), 239–248.
J. S. Wilson and E. I. Zelmanov, Identities for Lie algebras of pro-\(p\) groups, J. Pure Appl. Algebra 81 (1992), 103–109.
E. I. Zelmanov, On the restricted Burnside problem, In: Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), 395–402, Math. Soc. Japan, Tokyo, 1991.
E. I. Zelmanov, Lie algebras and torsion groups with identity, J. Comb. Algebra 1 (2017), 289–340.
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