Kourovka Notebook Problem 9.37

Definitions and mathematical statement

A topological group $G$ is compactly generated if there exists a compact subset $K\subseteq G$ such that $G=\langle K\rangle$, where $\langle K\rangle$ denotes the subgroup generated by $K$.

The supplied problem asks whether every compactly generated, inductively prosoluble, locally compact group is prosoluble. Its logical form is \[ \forall G\;\left[\left(G\text{ is a locally compact topological group}\right)\land\left(\exists K\subseteq G\text{ compact}:G=\langle K\rangle\right)\land\mathsf{IndProSol}(G)\;\Longrightarrow\;\mathsf{ProSol}(G)\right]. \] Here $\mathsf{IndProSol}(G)$ and $\mathsf{ProSol}(G)$ are placeholders for the source's meanings of "inductively prosoluble" and "prosoluble", respectively. The supplied record does not define these terms or specify the convention concerning Hausdorffness for locally compact groups. Consequently, this is a conditional transcription of the intended implication; a fully specified mathematical statement requires those source conventions. In particular, no interpretation in terms of inverse limits or directed unions is imposed.

Short English statement

Must every compactly generated, inductively prosoluble, locally compact group be prosoluble? The meanings of the two solubility conditions require clarification from the source. The supplied registry labels the problem open; that label is not independently verified here.

Sources

• Local UnsolvedMath registry, permanent ID $700570$, source code catalogue entry, title "Kourovka Notebook Problem 9.37". The supplied record attributes the problem to Yu. N. Mukhin and dates its proposal to 1984.
• The Kourovka Notebook (Unsolved Problems in Group Theory), original problem 9.37, 9th issue (1984); supplied reference edition: 21st issue edition (2026), arXiv:1401.0300v48. The registry describes the collection as edited by E. I. Khukhro and V. D. Mazurov.
• The supplied registry status is "open". The record supplies no dated literature-verification assessment; its bibliographic information and status have not been independently checked for this draft.