Kourovka Notebook Problem 19.68

Definitions and mathematical statement

Let $G$ be a finite abelian group of order $n\geq 1$. Interpret its rank as its minimum number of generators, \[ r=d(G):=\min\{\lvert S\rvert:S\subseteq G,\ \langle S\rangle=G\}, \] with $d(1)=0$. Let $\Omega$ be a nonempty finite set and let $K\leq\operatorname{Sym}(\Omega)$. A subgroup $H\leq K$ is regular on $\Omega$ if, for every $\alpha,\beta\in\Omega$, there exists exactly one $h\in H$ such that $h(\alpha)=\beta$. Define \[ b_G(K):=\left\lvert\{H\leq K:H\cong G,\ H\text{ is regular on }\Omega\}/\!\sim_K\right\rvert, \] where $H\sim_K H'$ means $H'=kHk^{-1}$ for some $k\in K$.

For a precise provisional interpretation of the referenced notation, use the standard $2$-closure \[ K^{(2)}:=\left\{\pi\in\operatorname{Sym}(\Omega): (\pi(\alpha),\pi(\beta))\in K\cdot(\alpha,\beta) \text{ for every }(\alpha,\beta)\in\Omega^2\right\}, \] where \[ K\cdot(\alpha,\beta)=\{(k(\alpha),k(\beta)):k\in K\}. \] Thus $K^{(2)}$ consists of the permutations preserving every orbit of the diagonal action of $K$ on ordered pairs, including pairs with equal coordinates. The source refers to Archive 19.67 for this definition, but that archive entry is not supplied; agreement with its convention requires review.

Question. Does there exist a function \[ f:\mathbb Z_{\geq 0}\longrightarrow\mathbb R_{\geq 0} \] such that, for every finite abelian group $G$ of order $n\geq 1$ and rank $r=d(G)$, every nonempty finite permutation domain $\Omega$, and every $K\leq\operatorname{Sym}(\Omega)$ satisfying $K^{(2)}=K$, one has \[ b_G(K)\leq n^{f(r)}\,? \] The same function must work for all these choices. The source does not specify the degree of $K$. A regular subgroup isomorphic to $G$ forces $\lvert\Omega\rvert=n$; otherwise $b_G(K)=0$. Consequently, only degree $n$ contributes a nonzero count. The source also leaves the meaning of rank and the domain and codomain of $f$ implicit; the conventions above make the proposed statement explicit and require source confirmation.

Short English statement

Is the number of $K$-conjugacy classes of regular subgroups isomorphic to a finite abelian group $G$ bounded by a polynomial in $\lvert G\rvert$, whose exponent depends only on the minimum number of generators of $G$, uniformly over all $2$-closed permutation groups $K$? The referenced definition of $2$-closure and the implicit rank convention require confirmation.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $701616$, exact source problem code catalogue entry, title Kourovka Notebook Problem 19.68; upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• Attributed in the record to I. Ponomarenko, proposed in 2018. The supplied source is The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), Problem 19.68, as represented in the 21st issue edition (2026), identified as arXiv:1401.0300v48. The collection description names E. I. Khukhro and V. D. Mazurov as editors.
• The statement refers to Archive 19.67 for the definition of $K^{(2)}$. Its contents are not included in the supplied record. No source was independently consulted for this draft.
• The registry labels the problem open and records an update timestamp of 2026-10-16T14:00:00Z. This is supplied registry metadata, not an independently verified assessment of the literature. No separate literature_checked_at date or literature-source links are supplied.