Kourovka Notebook Problem 17.10

Definitions and mathematical statement

Let $G$ be an arbitrary group, with identity $1$. Use the commutator convention $[x,y]=x^{-1}y^{-1}xy$, and define iterated commutators recursively by \[ [x,{}_1 y]=[x,y],\qquad [x,{}_{n+1}y]=[[x,{}_n y],y]\quad(n\geq 1). \] An element $a\in G$ is a right $4$-Engel element if \[ \forall g\in G,\qquad [a,{}_4 g]=1. \] A group is locally nilpotent if each of its finitely generated subgroups is nilpotent. The Hirsch--Plotkin radical $\operatorname{HP}(G)$ is the largest normal locally nilpotent subgroup of $G$.

The question is whether \[ \exists G\ \exists a\in G\quad \left(\left(\forall g\in G,\ [a,{}_4 g]=1\right) \ \land\ a\notin\operatorname{HP}(G)\right). \] No finiteness assumption or other restriction on $G$ is imposed in the supplied statement.

Short English statement

Does there exist a group with a right $4$-Engel element lying outside its Hirsch--Plotkin radical, that is, outside its largest normal locally nilpotent subgroup?

Sources

• Local UnsolvedMath registry, supplied record with permanent ID $701312$, exact source code catalogue entry, titled Kourovka Notebook Problem 17.10. The registry labels the problem open; this status is supplied metadata and has not been independently verified.
• The Kourovka Notebook (Unsolved Problems in Group Theory), Problem 17.10, proposed by A. Abdollahi in 2010. The supplied background identifies the 17th issue (2010) and the 21st issue edition (2026), arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors.