Kourovka Notebook Problem 8.58

Definitions and mathematical statement

Let $G$ be a topological group with identity $1$. Say that $G$ is locally nilpotent if every subgroup generated by finitely many elements is nilpotent, where a group $H$ is nilpotent if its lower central series $\gamma_1(H)=H$, $\gamma_{k+1}(H)=[\gamma_k(H),H]$ reaches $\{1\}$ after finitely many steps. Say that $G$ is torsion-free if, for every $g\in G$ and every integer $n\geq 1$, $g^n=1$ implies $g=1$. Here the source's phrase "without elements of finite order" is understood to exclude nonidentity elements of finite order.

A subgroup $N\leq G$ is isolated in $G$ if \[ \forall g\in G\;\forall n\in{\mathbb{Z}}_{\geq 1},\qquad g^n\in N\Longrightarrow g\in N. \] Closedness refers to the topology of $G$, and normality means $gNg^{-1}=N$ for every $g\in G$.

The literal source question asks whether every locally compact, locally nilpotent, torsion-free, nonabelian topological group $G$ has a subgroup $N$ satisfying \[ N\trianglelefteq G,\qquad N\text{ is closed},\qquad N\neq G,\qquad \forall g\in G\;\forall n\in{\mathbb{Z}}_{\geq 1},\quad g^n\in N\Longrightarrow g\in N. \] The source does not require $N\neq\{1\}$. Whether a nontrivial subgroup was intended is unresolved; that additional requirement has not been inserted. The source also does not explicitly specify a Hausdorff convention for locally compact groups.

Short English statement

Must every locally compact, locally nilpotent, torsion-free, nonabelian group contain a proper closed normal subgroup that contains every element having a positive power in that subgroup? The source leaves unclear whether the subgroup must also be nontrivial and whether Hausdorffness is assumed.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $700507$, exact source problem code catalogue entry, title "Kourovka Notebook Problem 8.58"; upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• Attributed to V. M. Poletskikh, proposed in 1982. The supplied background identifies The Kourovka Notebook (Unsolved Problems in Group Theory), 8th issue (1982), Problem 8.58, and the 21st issue edition (2026), arXiv:1401.0300v48. The collection description names E. I. Khukhro and V. D. Mazurov as editors.
• The registry labels this problem "open". This is supplied source metadata, not an independently verified literature assessment. The cited Notebook text was not independently consulted for this draft.