Gunnar Traustason (University of Bath) – Left 3-Engel elements in groups

An element $x$ in a group $G$ is a left Engel element if for each $x ∈ G$ there exists a positive integer $n = n(x)$ such that $[[[g,x],x],··· ,x] = 1$ ($n$ times). If $n = n(x)$ can be chosen independently of $x$, then we say that $x$ is a left $n$-Engel element. There are some connections to groups of prime power exponent and for example, every element in a group of exponent 3 is a left 2-Engel element. Whereas it is easy to see that the normal closure of a left 2-Engel element is abelian, it is still an open question whether the normal closure of a left 3-Engel element is locally nilpotent. We will give some overview of this problem, focusing on advances in recent years.