Abstract
An Engel sink of an element g of a group G is a set \({\cal E}(g)\) such that for every x ∈ G all sufficiently long commutators [⋯[[x, g], g],…, g] belong to \({\cal E}(g)\). (Thus, g is an Engel element precisely when we can choose \({\cal E}(g) = \{ 1\} \).) It is proved that if a profinite group G admits an elementary abelian group of automorphisms A of coprime order q2 for a prime q such that for each a ∈ A {1} every element of the centralizer CG(a) has a countable (or finite) Engel sink, then G has a finite normal subgroup N such that G/N is locally nilpotent.
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References
C. Acciarri, E. I. Khukhro and P. Shumyatsky, Profinite groups with an automorphism whose fixed points are right Engel, Proceedings of the American Mathematical Society 147 (2019), 3691–3703.
C. Acciarri and P. Shumyatsky, Profinite groups and the fixed points of coprime automorphisms, Journal of Algebra 452 (2016), 188–195.
C. Acciarri, P. Shumyatsky and D. S. Silveira, On groups with automorphisms whose fixed points are Engel, Annali di Matematica Pura ed Applicata 197 (2018), 307–316.
C. Acciarri, P. Shumyatsky and D. Silveira, Engel sinks of fixed points in finite groups, Journal of Pure and Applied Algebra 223 (2019), 4592–4601.
C. Acciarri and D. Silveira, Profinite groups and centralizers of coprime automorphisms whose elements are Engel, Journal of Group Theory 21 (2018), 485–509.
C. Acciarri and D. Silveira, Engel-like conditions in fixed points of automorphisms of profinite groups, Annali di Matematica Pura ed Applicata 199 (2020), 187–197.
Yu. A. Bahturin and M. V. Zaicev, Identities of graded algebras, Journal of Algebra 205 (1998), 1–12.
N. Bourbaki, Lie Groups and Lie Algebras. Chapters 1–3, Elements of Mathematics, Springer, Berlin, 1989.
J. D. Dixon, M. P. F. du Sautoy, A. Mann and D. Segal, Analytic Pro-p Groups, Cambridge Studies in Advanced Mathematics, Vol. 61, Cambridge University Press, Cambridge, 1999.
D. Gorenstein, Finite Groups, Chelsea, New York, 1980.
B. Huppert, Endliche Gruppen. I, Die Grundlehren der mathematischen Wissenschaften, Vol. 134, Springer, Berlin-New York, 1967.
J. L. Kelley, General Topology, Graduate Texts in Mathematics, Vol. 27, Springer, New York, 1975.
E. I. Khukhro and P. Shumyatsky, Bounding the exponent of a finite group with automorphisms, Journal of Algebra 212 (1999), 363–374.
E. I. Khukhro and P. Shumyatsky, Almost Engel finite and profinite groups, International Journal of Algebra and Computation 26 (2016), 973–983.
E. I. Khukhro and P. Shumyatsky, Almost Engel compact groups, Journal of Algebra 500 (2018), 439–456.
E. I. Khukhro and P. Shumyatsky, Finite groups with Engel sinks of bounded rank, Glasgow Mathematical Journal 60 (2018), 695–701.
E. I. Khukhro and P. Shumyatsky, Compact groups all elements of which are almost right Engel, Quarterly Journal of Mathematics 70 (2019), 879–893.
E. I. Khukhro and P. Shumyatsky, Compact groups with countable Engel sinks, Bulletin of Mathematical Sciences, Article no. 2050015.
E. I. Khukhro, P. Shumyatsky and G. Traustason, Right Engel-type subgroups and length parameters of finite groups, Journal of the Australian Mathematical Society 109 (2020), 340–350.
D. E. Knuth, The Art of Computer Programming: Volume 1: Fundamental Algorithms, Addison-Wesley, Reading, MA, 1997.
M. Lazard, Groupes analytiques p-adiques, Institut des Hautes Études Scientifiques. Publications Mathématiques 26 (1965), 389–603.
A. Lubotzky and A. Mann, Powerful p-groups. II: p-adic analytic groups, Journal of Algebra 105 (1987), 506–515.
L. Ribes and P. Zalesskii, Profinite Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 40, Springer, Berlin, 2010.
D. J. S. Robinson, A Course in the Theory of Groups, Graduate Texts in Mathematics, Vol. 80, Springer, New York, 1996.
A. Shalev, Polynomial identities in graded group rings, restricted Lie algebras and p-adic analytic groups, Transactions of the American Mathematical Society 337 (1993), 451–462.
J. N. Ward, On groups admitting a noncyclic abelian automorphism group, Bulletin of the Australian Mathematical Society 9 (1973), 363–366.
J. S. Wilson, Profinite Groups, London Mathematical Society Monographs, Vol. 19 Clarendon Press, Oxford University Press, New York, 1998.
J. S. Wilson and E. I. Zelmanov, Identities for Lie algebras of pro-p groups, Journal of Pure and Applied Algebra 81 (1992), 103–109.
E. Zelmanov, Nil Rings and Periodic Groups, KMS Lecture Notes in Mathematics, Korean Mathematical Society, Seoul, 1992.
E. Zelmanov, Lie methods in the theory of nilpotent groups, in Groups’ 93 Galaway/St Andrews, Vol. 2, London Mathematical Society Lecture Notes Series, Vol. 212, Cambridge University Press, Cambridge, 1995, pp. 567–585.
E. Zelmanov, Lie algebras and torsion groups with identity, Journal of Combinatorial Algebra 1 (2017), 289–340.
Acknowledgements
The first author was supported by the Mathematical Center in Akademgorodok, the agreement with Ministry of Science and High Education of the Russian Federation no. 075-15-2019-1613. The second author was supported by FAPDF and CNPq-Brazil.
The authors are grateful to the referee for the comments, which helped to improve the presentation.
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Khukhro, E.I., Shumyatsky, P. On profinite groups with automorphisms whose fixed points have countable Engel sinks. Isr. J. Math. 247, 303–330 (2022). https://doi.org/10.1007/s11856-021-2267-1
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DOI: https://doi.org/10.1007/s11856-021-2267-1