Kourovka Notebook Problem 13.15

Definitions and mathematical statement

For a set $X$ with $|X|\geq 2$, let $F(X)$ be the free group on $X$. Define its lower central series by $\gamma_1(F(X))=F(X)$ and $\gamma_{k+1}(F(X))=[\gamma_k(F(X)),F(X)]$. The free nilpotent group of class $3$ on $X$ is \[ N_X=F(X)/\gamma_4(F(X)). \] The source does not specify the rank $|X|$, whether it is finite, or the intended quantifier over ranks; this qualification remains unresolved.

A group $G$ is torsion-free if, for every $g\in G$ and every integer $n\geq 1$, the equality $g^n=1$ implies $g=1$. A group quasiidentity is a universally quantified sentence \[ \forall x_1\cdots\forall x_n\;\left(\bigwedge_{j=1}^{m}u_j(x_1,\ldots,x_n)=1\ \Longrightarrow\ v(x_1,\ldots,x_n)=1\right), \] where $n\geq 1$, $m\geq 0$, and $u_1,\ldots,u_m,v$ are group words; the empty conjunction is interpreted as true. Write $\operatorname{QI}(N_X)$ for all quasiidentities satisfied by $N_X$.

Interpreting a basis relative to torsion-free groups in the semantic sense, a set $\Sigma\subseteq\operatorname{QI}(N_X)$ is such a basis if, for every torsion-free group $G$, \[ G\models\Sigma\quad\Longleftrightarrow\quad G\models\operatorname{QI}(N_X). \] It is independent if, for every $\sigma\in\Sigma$, there exists a torsion-free group $G_\sigma$ such that \[ G_\sigma\models\Sigma\setminus\{\sigma\}, \qquad G_\sigma\not\models\sigma. \] The question is whether $N_X$ possesses an independent basis of quasiidentities relative to torsion-free groups. No finiteness restriction on the basis is stated. The displayed formulation is parameterized by $X$; it does not resolve the source's missing rank quantifier.

Short English statement

Does a free nonabelian nilpotent group of class $3$ admit a set of quasiidentities that determines all its quasiidentities among torsion-free groups, with every member indispensable within that class? The source leaves the rank and its quantification unspecified.

Sources

• Local UnsolvedMath registry: permanent ID $700933$, source problem code catalogue entry, title Kourovka Notebook Problem 13.15; supplied upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• Supplied attribution: A. I. Budkin, proposed in 1995; The Kourovka Notebook (Unsolved Problems in Group Theory), 13th issue (1995), Problem 13.15. The supplied background also identifies the 21st issue edition (2026), arXiv:1401.0300v48. These bibliographic details are supplied source information and have not been independently verified.
• The registry labels the problem open. This is a supplied status label, not an independently verified mathematical or literature assessment.