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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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