Kourovka Notebook Problem 15.23

Definitions and mathematical statement

The supplied record does not define barely transitive. The formulation below uses the convention that this means a faithful transitive action on an infinite set in which every proper subgroup has only finite orbits. This interpretation, including the infinitude requirement, requires confirmation against the cited source.

Let $\Omega$ be an infinite set and let $G\leq\operatorname{Sym}(\Omega)$. The action is transitive if for all $\alpha,\beta\in\Omega$ there exists $g\in G$ with $g\alpha=\beta$. A block is a nonempty subset $B\subseteq\Omega$ such that, for every $g\in G$, either $gB=B$ or $gB\cap B=\varnothing$. As defined in the supplied statement, the action is totally imprimitive if \[ \forall S\subseteq\Omega\text{ finite},\quad \exists B\subseteq\Omega\text{ a finite block with }S\subseteq B. \] Under the convention stated above, the action is barely transitive if it is transitive and \[ \forall H<G\;\forall\alpha\in\Omega,\qquad |H\alpha|<\infty, \qquad H\alpha=\{h\alpha:h\in H\}. \] A group is locally finite if every subgroup generated by a finite subset is finite.

Question (V. V. Belyaev, 2002; Kourovka Notebook Problem 15.23). Does there exist an infinite set $\Omega$ and a transitive group $G\leq\operatorname{Sym}(\Omega)$ whose action is totally imprimitive and barely transitive, but for which \[ \exists F\subseteq G\text{ finite},\qquad |\langle F\rangle|=\infty? \] The supplied registry labels this problem open; that label is not an independently verified assessment.

Short English statement

Under the stated interpretation of barely transitive, the question asks whether an infinite transitive permutation group can have every finite set of points contained in a finite block and every proper subgroup acting with finite orbits, while possessing an infinite finitely generated subgroup. The interpretation of barely transitive requires source confirmation.

Sources

• Local UnsolvedMath registry, permanent ID $701111$, exact source code catalogue entry, title Kourovka Notebook Problem 15.23; supplied upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82. The supplied status is open.
• The Kourovka Notebook (Unsolved Problems in Group Theory), 15th issue (2002), Problem 15.23, proposed by V. V. Belyaev, as attributed in the supplied record.
• The supplied background also identifies the 21st issue edition (2026), arXiv:1401.0300v48. The collection description credits E. I. Khukhro and V. D. Mazurov as editors of Issues 1--21 (1965--2026). These bibliographic details are supplied attribution and have not been independently checked.