Left 3-Engel Elements in Locally Finite p-groups

Abstract: Engel Theory is of significant interest in group theory as there is an unmistakable correlation between Engel and Burnside problems. In this talk we first introduce Engel elements and Engel groups and in particular we expand our knowledge on locally finite Engel groups. It is important to know that M. Newell proved that if $x$ is a right $3$-Engel element in a group $G$ then $x$ lies in $HP(G)$ (Hirsch-Plotkin radical) and in fact he proved the stronger result that the normal closure of $x$ is nilpotent of class at most $3$. The natural question arises whether the analogous result holds for left $3$-Engel elements. We will give various examples of locally finite p-groups $G$ containing a left $3$-Engel element $x$ whose normal closure is not nilpotent. Lastly, we will discuss the open problem of whether or not a left $3$-Engel element always lies in the $HP(G)$. (This is joint work with Gunnar Traustason and Marialaura Noce)

Mathematics

Audience: researchers in the topic

Greek Algebra & Number Theory Seminar