Kourovka Notebook Problem 17.11

Definitions and mathematical statement

Let $G$ be a group, with identity $1$. Use the commutator convention $[x,y]=x^{-1}y^{-1}xy$. An element $a\in G$ is a left $3$-Engel element if \[ [[[x,a],a],a]=1\qquad\text{for every }x\in G. \] A group is locally nilpotent if each of its finitely generated subgroups is nilpotent. Here a group $H$ is nilpotent if its lower central series, defined by $\gamma_1(H)=H$ and $\gamma_{n+1}(H)=[\gamma_n(H),H]$, reaches the trivial subgroup after finitely many steps; $[U,V]$ denotes the subgroup generated by all commutators $[u,v]$ with $u\in U$ and $v\in V$. The Hirsch--Plotkin radical $\operatorname{HP}(G)$ is the largest normal locally nilpotent subgroup of $G$.

Problem (A. Abdollahi, 2010). Does there exist a group $G$ and an element $a\in G$ such that \[ \left(\forall x\in G,\; [[[x,a],a],a]=1\right) \quad\text{and}\quad a\notin\operatorname{HP}(G)? \] No finiteness assumption on $G$ is imposed in the supplied statement.

Short English statement

Does some group contain a left $3$-Engel element outside its Hirsch--Plotkin radical, that is, outside its largest normal locally nilpotent subgroup?

Sources

• Local UnsolvedMath registry: permanent ID $701313$, source problem code catalogue entry, title "Kourovka Notebook Problem 17.11"; supplied upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• The supplied background attributes Problem 17.11 to A. Abdollahi, proposed in 2010, in The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), and cites the 21st issue edition (2026), arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors.
• The registry labels the problem open. This is a supplied source classification, not an independently verified assessment of the literature. The cited editions and arXiv version have not been independently checked for this draft.