19.83 (2018)

An element $g$ of a group $G$ is almost Engel if there is a finite set $\mathcal{E}(g)$ such that for every $x \in G$ all sufficiently long commutators $[x, {}_n g]$ belong to $\mathcal{E}(g)$, that is, for every $x \in G$ there is a positive integer $n(x, g)$ such that $[x, {}_n g] \in \mathcal{E}(g)$ whenever $n \geqslant n(x, g)$. By a linear group we understand a subgroup of $\text{GL}(m, F)$ for some field $F$ and a positive integer $m$. Is the set of almost Engel elements in a linear group always a subgroup?