5 problems

Let $G$ be an unbounded quasi-Engel group, meaning that for every $x,y\in G$ there is an integer $n=n(x,y)$ such that ${}^w\!u_n(x,y)=1$, for the fixed initial word $w$. A group is…

Let $G$ be a quasi-Engel group, meaning that for the fixed initial word $w$ there is an integer $n$ such that the identity ${}^w\!u_n(x,y)\equiv 1$ holds in $G$. A group is residua…

Let $G \times H$ be a group, let $(u,v) \in G \times H$, and let $(u,v)^{G \times H}$ denote the normal closure of $(u,v)$ in $G \times H$. The associated subgroups…

Let $w$ be a group-word and let $n$ be a positive integer. For a group $G$, write $w(G)$ for the verbal subgroup generated by all values of $w$ in $G$. An element is $n$-Engel if,…

Let $G$ be an Engel group, meaning that there is an integer $n$ such that … for all $x,y\in G$. Engel-group conjecture. There exists an Engel group that is not locally nilpotent. T…