Kourovka Notebook Problem 8.60

Definitions and mathematical statement

Let $G$ be a locally compact topological group. Say that $G$ is covered by compact subgroups if \[ \forall g\in G\;\exists K\leq G\quad(K\text{ is compact and }g\in K). \] The classification problem in the supplied record is to describe all such $G$ that are primary and locally soluble and satisfy \[ \forall A\leq G,\qquad \left(A\text{ is closed in }G\text{ and abelian}\right) \Longrightarrow A\text{ has finite rank}. \] The finite rank condition is imposed separately on each closed abelian subgroup; no bound uniform over all such subgroups is stated.

The terms "primary", "locally soluble", and "finite rank" must retain their intended source meanings, which the supplied excerpt does not define. Under the usual abstract interpretation, locally soluble means that every finitely generated subgroup is soluble: for each such subgroup $H$, there exists $d\geq0$ with $H^{(d)}=\{1\}$, where $H^{(0)}=H$ and $H^{(i+1)}=[H^{(i)},H^{(i)}]$. This interpretation requires confirmation in the topological setting.

For comparison only, finite abstract Prüfer rank for an abelian group $A$ means that there exists an integer $r_A\geq0$ such that every finitely generated subgroup of $A$ can be generated by at most $r_A$ elements. The record does not establish that this is the intended rank, rather than a topological rank convention. Likewise, it does not establish whether "primary" requires a common prime $p$ with every element of finite $p$-power order, or uses a topological notion of $p$-primary group. These alternatives are not identified here. Consequently, this is a faithful classification template, with unresolved terminology, rather than a uniquely specified expanded formulation. The record also does not specify the equivalence relation for the requested classification.

Short English statement

Describe the locally compact primary locally soluble groups in which every element belongs to a compact subgroup and every closed abelian subgroup has finite rank. The meanings of primary, local solubility in this topological setting, and rank require source clarification; no uniform rank bound is stated.

Sources

• Local UnsolvedMath registry, permanent ID $700509$, exact source problem code catalogue entry, title "Kourovka Notebook Problem 8.60"; supplied upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• Supplied attribution: V. M. Poletskikh and V. S. Charin; proposed in 1982.
• Supplied source: The Kourovka Notebook (Unsolved Problems in Group Theory), 8th issue (1982), Problem 8.60; also identified with the 21st issue edition (2026), arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors.
• The local record labels the problem "open" and gives an update timestamp of 2026-10-16T14:00:00Z. This is supplied registry metadata, not an independently verified assessment of the literature or mathematical status. The referenced source has not been independently inspected for this draft.