# Charlotte Scott Centre for Algebra

School of Mathematics & Physics, University of Lincoln

# New paper accepted by Israel Journal of Mathematics

New paper by Evgeny Khukhro (Univ. of Lincoln) and Pavel Shumyatsky (Univ. of Brasilia)On profinite groups with automorphisms whose fixed points have countable Engel sinkss” has been accepted for publication in the Israel Journal of Mathematics. The work was supported by Mathematical Center in Akademgorodok, FAPDF and CNPq-Brazil, and stems from the collaboration with University of Brasilia.

Abstract: An Engel sink of an element $g$ of a group $G$ is a set ${\mathscr E}(g)$ such that for every $x\in G$ all sufficiently long commutators $[\dots [[x,g],g],\dots ,g]$ belong to ${\mathscr E}(g)$. (Thus, $g$ is an Engel element precisely when we can choose ${\mathscr E}(g)={ 1}$.) It is proved that if a profinite group $G$ admits an elementary abelian group of automorphisms $A$ of coprime order $q^2$ for a prime $q$ such that for each $a\in A\setminus{1}$ every element of the centralizer $C_G(a)$ 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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This entry was posted on March 12, 2021 by in grants, New publications, research.