# Charlotte Scott Centre for Algebra

School of Mathematics & Physics, University of Lincoln

# New paper accepted by Monatshefte für Mathematik

New paper by Evgeny Khukhro (Univ. of Lincoln) and Pavel Shumyatsky (Univ. of Brasilia) “Profinite groups with an automorphism of prime order whose fixed points have finite Engel sinks” has been accepted for publication in the Monatshefte für Mathematik. The work was supported by FAPDF and CNPq-Brazil, and stems from the collaboration with University of Brasilia.

Abstract: A right Engel sink of an element $g$ of a group $G$ is a set ${\mathscr R}(g)$ such that for every $x\in G$ all sufficiently long commutators $[\dots [[g,x],x],\dots ,x]$ belong to ${\mathscr R}(g)$. (Thus, $g$ is a right Engel element precisely when we can choose ${\mathscr R}(g)={ 1}$.) We prove that if a profinite group $G$ admits a coprime automorphism $\varphi$ of prime order such that every fixed point of $\varphi$ has a finite right Engel sink, then $G$ has an open locally nilpotent subgroup.

A left 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 a left Engel element precisely when we can choose ${\mathscr E}(g)={ 1}$.) We prove that if a profinite group $G$ admits a coprime automorphism $\varphi$ of prime order such that every fixed point of $\varphi$ has a finite left Engel sink, then $G$ has an open pronilpotent-by-nilpotent subgroup.

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This entry was posted on April 20, 2021 by in New publications, research.