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.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

This site uses Akismet to reduce spam. Learn how your comment data is processed.

Information

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

Blog Stats

  • 26,845 hits

Archives

%d bloggers like this: