Kourovka Notebook Problem 3.38

Definitions and mathematical statement

Let $G$ be a group with identity $e$, equipped with a topology $\tau$ such that multiplication $G\times G\to G$, with the product topology on $G\times G$, and inversion $G\to G$ are continuous. A subgroup $H\leq G$ is closed if $G\setminus H\in\tau$, and proper if $H\ne G$.

Describe the topological groups $(G,\tau)$ satisfying \[ \forall H\leq G,\qquad H\text{ is closed in }G\ \Longrightarrow\ H=G. \] The quantifier includes the identity subgroup $\{e\}$; the source does not restrict the condition to nontrivial subgroups or exclude the trivial group.

The source does not specify whether topological groups must satisfy the Hausdorff axiom or another separation axiom. The continuity conditions above state the common underlying structure but do not resolve that convention. The intended separation hypotheses must therefore be checked before this classification request is regarded as fully specified.

Short English statement

Describe all topological groups whose only closed subgroup is the whole group. The source leaves the separation convention unspecified, and this qualification requires review.

Sources

• Local UnsolvedMath registry, supplied record with permanent ID $700187$, exact source code catalogue entry, titled Kourovka Notebook Problem 3.38. The registry labels the problem open; this is supplied source metadata, not an independently verified mathematical or literature assessment.
• The Kourovka Notebook (Unsolved Problems in Group Theory), Problem 3.38, proposed by Yu. N. Mukhin in the third issue (1969). The supplied record identifies the twenty-first-issue edition (2026), arXiv:1401.0300v48, as its source. The collection description names E. I. Khukhro and V. D. Mazurov as editors. This bibliographic attribution is supplied by the registry and has not been independently checked.