Kourovka Notebook Problem 17.13

Definitions and mathematical statement

Let $p$ be a prime and let $\Omega$ be an infinite set. For a permutation $g$ of $\Omega$, define its support by \[ \operatorname{supp}(g)=\{\omega\in\Omega:g(\omega)\ne\omega\}. \] Write $\operatorname{Sym}_{\mathrm{fin}}(\Omega)$ for the group of permutations with finite support. A subgroup $G\leq\operatorname{Sym}_{\mathrm{fin}}(\Omega)$ is a $p$-group if every element has order a power of $p$. A nonempty subset $B\subseteq\Omega$ is a block for $G$ if, for every $g\in G$, either $g(B)=B$ or $g(B)\cap B=\varnothing$.

For a group $H$ and $h\in H$, write \[ h^H=\{x^{-1}hx:x\in H\}. \] The group $H$ is an $FC$-group if $h^H$ is finite for every $h\in H$. A minimal non-$FC$-subgroup of $G$ means a subgroup $H\leq G$ that is not an $FC$-group and whose every proper subgroup is an $FC$-group. Minimality here is with respect to subgroup inclusion.

The source also requires the permutation action of $G$ on $\Omega$ to be totally imprimitive. The supplied record does not define this term. It is retained as an unresolved source qualification; the formulation below depends on its intended definition.

Question (A. O. Asar, Kourovka Notebook Problem 17.13). For every prime $p$, every infinite set $\Omega$, and every totally imprimitive $p$-group $G\leq\operatorname{Sym}_{\mathrm{fin}}(\Omega)$, suppose that the support of every nontrivial cycle in the disjoint cycle decomposition of every $g\in G$ is a block for $G$. Must there exist $H\leq G$ such that \[ \left(\exists h\in H:\ |h^H|=\infty\right) \quad\text{and}\quad \left(\forall K<H\ \forall k\in K:\ |k^K|<\infty\right)? \] Here $K<H$ means that $K$ is a proper subgroup of $H$. If fixed points are included as one-point cycles, their underlying singleton sets are automatically blocks, so they impose no additional condition.

Short English statement

Asar asks whether every totally imprimitive group of finitary permutations of prime-power orders, satisfying the stated block condition on all cycle supports, contains a subgroup that is not an $FC$-group although every proper subgroup is an $FC$-group. The intended definition of "totally imprimitive" remains to be confirmed from the source.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $701315$, exact source problem code catalogue entry, title Kourovka Notebook Problem 17.13, upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82. The registry labels the problem open; this is a supplied status, not an independently verified mathematical or literature assessment.
• Attributed source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), Problem 17.13, proposed by A. O. Asar. The supplied background also identifies the 21st issue edition (2026), arXiv:1401.0300v48.
• The supplied collection description credits E. I. Khukhro and V. D. Mazurov as editors of the Kourovka Notebook collection covering Issues 1--21 (1965--2026). These bibliographic details are supplied attribution and have not been independently checked.