Kourovka Notebook Problem 8.59

Definitions and mathematical statement

Let $G$ be a locally compact topological group. For this draft, local compactness is understood with the usual Hausdorff convention. Interpret locally nilpotent algebraically: every subgroup generated by finitely many elements of $G$ is nilpotent. Here a group $H$ is nilpotent if its lower central series, defined by $\gamma_1(H)=H$ and $\gamma_{i+1}(H)=[\gamma_i(H),H]$, reaches the trivial subgroup after finitely many steps; $[A,B]$ denotes the subgroup generated by commutators $a^{-1}b^{-1}ab$, with $a\in A$ and $b\in B$.

Assume that every proper closed normal subgroup of $G$ is compact; explicitly, \[ \forall N\leq G,\qquad \left(N\trianglelefteq G\ \land\ N\text{ is closed}\ \land\ N\ne G\right) \Longrightarrow N\text{ is compact}. \] Must there exist a subgroup $K\leq G$ such that \[ K\trianglelefteq G,\qquad K\text{ is open in }G,\qquad K\text{ is compact}? \] The desired subgroup $K$ is not required to be proper. All topological properties refer to the given topology on $G$. The supplied record does not explicitly specify the Hausdorff convention or define local nilpotence; the conventions stated above require confirmation against the source.

Short English statement

Must every locally compact, locally nilpotent group whose proper closed normal subgroups are all compact contain a compact open normal subgroup? The requested subgroup may be the whole group. This draft uses the Hausdorff convention and algebraic local nilpotence, subject to source review.

Sources

• Local UnsolvedMath registry, permanent ID $700508$, exact source code catalogue entry, title Kourovka Notebook Problem 8.59. The supplied record labels the problem open; this label is not an independently verified mathematical or literature assessment.
• The Kourovka Notebook (Unsolved Problems in Group Theory), original problem 8.59, 8th issue (1982); supplied edition reference: 21st issue edition (2026), arXiv:1401.0300v48. Attribution: V. M. Poletskikh and I. V. Protasov, proposed in 1982. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors of Issues 1--21 (1965--2026). These bibliographic details are supplied attribution and have not been independently verified.