Kourovka Notebook Problem 12.68

Definitions and mathematical statement

Let $G$ be a locally compact abelian topological group, written additively. For subgroups $H,K\leq G$, their algebraically generated subgroup is \[ \langle H\cup K\rangle=H+K=\{h+k:h\in H,\ k\in K\}, \] whereas their generated closed subgroup is $\overline{H+K}^{\,G}$, with closure taken in $G$. These need not be the same subgroup.

The source asks for a characterization of those $G$ in which every pair of closed subgroups of finite rank generates a subgroup of finite rank. To display the quantifiers without supplying an unsupported definition, write $\operatorname{FR}(L)$ as a placeholder for the source's assertion that $L$ has finite rank, and write $J_G(H,K)$ for its intended generated subgroup. The requested property is \[ \forall H,K\leq G,\quad \left(H\text{ and }K\text{ are closed in }G \ \land\ \operatorname{FR}(H)\ \land\ \operatorname{FR}(K)\right) \Longrightarrow \operatorname{FR}\left(J_G(H,K)\right). \] The task is to describe all locally compact abelian groups satisfying this property. However, the supplied record does not define finite rank or explicitly resolve whether $J_G(H,K)=H+K$ or $J_G(H,K)=\overline{H+K}^{\,G}$. Thus the displayed property is a quantified formulation template, not a uniquely specified classification problem; these qualifications require resolution from the original source or human mathematical review.

Short English statement

Describe the locally compact abelian groups for which any two closed subgroups of finite rank generate a subgroup of finite rank. The supplied record leaves the meaning of finite rank unspecified and requires clarification about whether generation includes taking closure.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $700886$, exact source problem code catalogue entry, title Kourovka Notebook Problem 12.68. The supplied status is open; this is a registry label, not an independently verified mathematical assessment. Record updated 2026-10-16T14:00:00Z; upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• The Kourovka Notebook (Unsolved Problems in Group Theory), original Problem 12.68, 12th issue (1992); proposed by V. M. Poletskikh. The supplied background also identifies the 21st issue edition (2026), arXiv:1401.0300v48. The collection description names E. I. Khukhro and V. D. Mazurov as editors. These bibliographic details are supplied attribution and have not been independently checked.