Profinite groups with an automorphism of prime order whose fixed points have finite Engel sinks

DOI10.1007/S00605-021-01561-5arXiv2101.03404MaRDI QIDQ6357921

Evgenii I. Khukhro, Pavel Shumyatsky

Publication date: 9 January 2021

Abstract: A right Engel sink of an element $g$ of a group $G$ is a set $mathscrR(g)$ such that for every $xinG$ all sufficiently long commutators $[...[[g,x],x],dots,x]$ belong to $mathscrR(g)$. (Thus, $g$ is a right Engel element precisely when we can choose $mathscrR(g)=1$.) 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 $mathscrE(g)$ such that for every $xinG$ all sufficiently long commutators $[...[[x,g],g],dots,g]$ belong to $mathscrE(g)$. (Thus, $g$ is a left Engel element precisely when we can choose $mathscrE(g)=1$.) 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.

Mathematics Subject Classification ID

Generalizations of solvable and nilpotent groups (20F19) Automorphisms of infinite groups (20E36) Engel conditions (20F45) Limits, profinite groups (20E18)

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