Khukhro, Evgeny and Shumyatsky, Pavel (2021) Profinite groups with an automorphism of prime order whose fixed points have finite Engel sinks. Monatshefte für Mathematik . ISSN 00269255
Full content URL: https://doi.org/10.1007/s00605021015615
Documents 


PDF
khushu205.pdf  Whole Document Restricted to Repository staff only 308kB  

PDF
KhukhroShumyatsky2021_Article_ProfiniteGroupsWithAnAutomorph.pdf  Whole Document Available under License Creative Commons Attribution 4.0 International. 350kB 
Item Type:  Article 

Item Status:  Live Archive 
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 $[...[[g,x],x],\dots ,x]$ 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 $[...[[x,g],g],\dots ,g]$ 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 pronilpotentbynilpotent subgroup.
Keywords:  Profinite group, Engel condition, Automorphism, locally nilpotent 

Subjects:  G Mathematical and Computer Sciences > G110 Pure Mathematics 
Divisions:  College of Science > School of Mathematics and Physics 
ID Code:  44640 
Deposited On:  05 May 2021 11:18 
Repository Staff Only: item control page