Kourovka Notebook Problem 19.83

Definitions and mathematical statement

For a group $G$, use the commutator convention $[a,b]=a^{-1}b^{-1}ab$, and define iterated commutators recursively by \[ [x,{}_1 g]=[x,g],\qquad [x,{}_{n+1}g]=[[x,{}_n g],g]\quad(n\geq 1). \] An element $g\in G$ is almost Engel if there exists a finite set $\mathscr E(g)\subseteq G$ such that \[ \forall x\in G\;\exists n(x,g)\in{\mathbb{Z}}_{\geq 1}\;\forall n\in{\mathbb{Z}}_{\geq 1},\qquad n\geq n(x,g)\ \Longrightarrow\ [x,{}_n g]\in\mathscr E(g). \] The finite set may depend on $g$, but must work for every $x$; the threshold may depend on both $x$ and $g$. Write \[ A(G)=\{g\in G:g\text{ is almost Engel}\}. \] A linear group means a subgroup of $\operatorname{GL}(m,F)$, the group of invertible $m\times m$ matrices over a field $F$, for some positive integer $m$.

Question (P. Shumyatsky, 2018). For every field $F$, every positive integer $m$, and every subgroup $G\leq\operatorname{GL}(m,F)$, is $A(G)$ a subgroup of $G$?

Short English statement

In an arbitrary linear group, must the almost Engel elements form a subgroup? Here an element is almost Engel when, for each starting element, its sufficiently long iterated commutators eventually stay in a finite set depending only on the almost Engel element.

Sources

• Local UnsolvedMath registry, supplied source record with permanent ID $701631$, exact source problem code catalogue entry, title Kourovka Notebook Problem 19.83.
• P. Shumyatsky, Problem 19.83, The Kourovka Notebook (Unsolved Problems in Group Theory). The supplied background identifies the 19th issue (2018) and the 21st issue edition (2026), with identifier arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors.
• The supplied registry labels this problem open. This is a supplied source classification, not an independently verified assessment of its mathematical or literature status.