Kourovka Notebook Problem 13.24

Definitions and mathematical statement

Let $G$ be a topological group with identity $e$, whose topology is not discrete. An ultrafilter on the underlying set of $G$ means a maximal proper filter of subsets of $G$. Such an ultrafilter $\mathcal U$ converges to $e$ if every neighbourhood of $e$ belongs to $\mathcal U$. Put \[ \mathscr U_e(G)=\{\mathcal U:\mathcal U\text{ is an ultrafilter on }G\text{ and }\mathcal U\to e\}. \] Assume that $\mathscr U_e(G)$ is finite. Must there exist a subgroup $H\leq G$ such that \[ H\text{ is open in }G,\qquad |H|\leq\aleph_0,\qquad h^2=e\quad\text{for every }h\in H? \] Here countable means finite or countably infinite. The condition on squares expresses exponent $2$: since $G$ is non-discrete, an open subgroup cannot be $\{e\}$. Ultrafilters are understood to include principal ultrafilters; in particular, the principal ultrafilter at $e$ is included in $\mathscr U_e(G)$.

The supplied statement does not specify a separation axiom for topological groups. Thus the intended convention, particularly whether Hausdorffness is required, remains unresolved; the formulation above records the stated hypotheses without adding that requirement.

The supplied source labels the general problem open. Its comment dated 2005 reports that the conclusion was proved when $G$ is countable, citing E. G. Zelenyuk, Mat. Studii 7 (1997), 139--144. This is a supplied historical qualification, not an independently verified result or current literature assessment.

Short English statement

Must every non-discrete topological group with only finitely many ultrafilters converging to its identity contain a countable open subgroup in which every element squares to the identity? The intended separation convention needs clarification. A supplied 2005 comment reports an affirmative result for countable groups.

Sources

• Local UnsolvedMath registry: permanent ID $700942$, exact source code catalogue entry, title Kourovka Notebook Problem 13.24. The supplied registry status is open; this label has not been independently verified.
• Attributed to E. G. Zelenyuk, proposed in 1995. Supplied source: The Kourovka Notebook (Unsolved Problems in Group Theory), 13th issue (1995), Problem 13.24; 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.
• Supplied comment dated 2005: the countable case is reported proved by E. G. Zelenyuk, Mat. Studii 7 (1997), 139--144. No article title or further bibliographic details are supplied. Neither this citation nor the reported result has been independently checked.