Kourovka Notebook Problem 14.26

Definitions and mathematical statement

Work in the language of groups. A quasi-identity is a universally quantified implication \[ \forall x_1,\ldots,x_r\quad \left(\bigwedge_{i=1}^{m}u_i(x_1,\ldots,x_r)=v_i(x_1,\ldots,x_r)\right) \Longrightarrow u(x_1,\ldots,x_r)=v(x_1,\ldots,x_r), \] where all displayed terms are group terms and the conjunction may be empty. A quasivariety of groups is a class axiomatized by quasi-identities.

Let $\mathcal N$ be the class of all torsion-free nilpotent groups, without a fixed bound on nilpotency class. Here torsion-free means that $g^n=1$, for $n\geq 1$, implies $g=1$. Nilpotence means that $\gamma_{c+1}(G)=\{1\}$ for some finite integer $c\geq 1$, where $\gamma_1(G)=G$ and $\gamma_{j+1}(G)=[\gamma_j(G),G]$. Let \[ \mathcal Q=\operatorname{Qvar}(\mathcal N) \] be the smallest quasivariety containing $\mathcal N$; equivalently, $\mathcal Q$ consists of the groups satisfying every quasi-identity valid in every member of $\mathcal N$.

Interpret the source's term "direct wreath product" in its usual unrestricted sense. For a group $G$, let $G^{{\mathbb{Z}}}=\prod_{n\in{\mathbb{Z}}}G$ be the group of all functions ${\mathbb{Z}}\to G$, with pointwise multiplication and no finite-support restriction. For $k\in{\mathbb{Z}}$, define $\tau_k(f)(n)=f(n-k)$. The direct wreath product with the infinite cyclic group is \[ G\wr{\mathbb{Z}}=G^{{\mathbb{Z}}}\rtimes_{\tau}{\mathbb{Z}}, \qquad (f,k)(h,\ell)=\left(f\,\tau_k(h),k+\ell\right). \] A quasivariety $\mathcal M$ is closed under direct ${\mathbb{Z}}$-wreath products if $G\wr{\mathbb{Z}}\in\mathcal M$ for every $G\in\mathcal M$.

The question proposed by A. I. Budkin is whether \[ \forall G\in\mathcal Q,\qquad G\wr{\mathbb{Z}}\in\mathcal Q. \] The supplied registry labels the problem open. Its accompanying discussion reports an explicitly unverified AI claim of an affirmative solution; this draft does not treat that claim as an established resolution.

Short English statement

Consider the quasivariety generated by all torsion-free nilpotent groups, allowing every finite nilpotency class. Must the unrestricted wreath product $G\wr{\mathbb{Z}}$ belong to this quasivariety whenever $G$ does? The supplied open label and the reported unverified affirmative AI claim are retained as source qualifications.

Sources

• Local UnsolvedMath registry, supplied complete record with permanent ID $701011$, exact source problem code catalogue entry, title "Kourovka Notebook Problem 14.26." The record attributes the problem to A. I. Budkin, proposed in 1999.
• The supplied source attribution is The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), Problem 14.26, as included in the 21st issue edition (2026), arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors of Issues 1--21 (1965--2026).
• The supplied discussion reports an explicitly "unverified AI solution" claiming an affirmative answer, attributed to M. Kinyon and J. Urban, Preprint, 2026, with the supplied link https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2026/09/kourovka-mathematics.pdf. No separate preprint title is supplied.
• The supplied status review is dated 2026-10-16, retains the canonical status "open," and classifies the evidence as unrefereed_claim. It says that the October 2026 21st issue, arXiv:1401.0300v48, marks the problem with $\#$ for the unverified AI solution claim. This is supplied dated curation, not an independently verified literature assessment.