Kourovka Notebook Problem 13.14

Definitions and mathematical statement

Work in the language of groups, with multiplication, inversion, and identity element $1$. Write $[x,y]=x^{-1}y^{-1}xy$. Let $\mathcal K$ be the class of all groups $G$ satisfying \[ \forall x,y,z\in G,\qquad [[x,y],z]=1, \] and, for every integer $n\geq 1$, \[ \forall x\in G,\qquad x^n=1\Longrightarrow x=1. \] Thus $\mathcal K$ consists of the torsion-free nilpotent groups of nilpotency class at most $2$, including abelian groups and the trivial group. No finiteness or finite-generation hypothesis is imposed.

A group quasi-identity is a universally quantified implication \[ \forall x_1,\ldots,x_r,\qquad \left(\bigwedge_{i=1}^{m}s_i(x_1,\ldots,x_r)=t_i(x_1,\ldots,x_r)\right) \Longrightarrow s(x_1,\ldots,x_r)=t(x_1,\ldots,x_r), \] where all displayed expressions are group terms and $m$ is finite and may be zero. A quasivariety of groups is the class of all groups satisfying some set of group quasi-identities. The phrase "quasivarieties of nilpotent torsion-free groups of nilpotency class at most $2$" is interpreted as quasivarieties of groups contained in $\mathcal K$.

Let $\mathcal L$ be the lattice of these quasivarieties, ordered by inclusion. Since $\mathcal K$ is itself axiomatized by quasi-identities, its lattice operations are \[ \mathcal U\wedge\mathcal V=\mathcal U\cap\mathcal V, \qquad \mathcal U\vee\mathcal V= \bigcap\{\mathcal W\in\mathcal L:\mathcal U\cup\mathcal V\subseteq\mathcal W\}. \] The join is therefore the smallest quasivariety containing both classes.

Question (A. I. Budkin). Is $\mathcal L$ distributive? Equivalently, does \[ \mathcal U\cap(\mathcal V\vee\mathcal W) =(\mathcal U\cap\mathcal V)\vee(\mathcal U\cap\mathcal W) \] hold for every $\mathcal U,\mathcal V,\mathcal W\in\mathcal L$?

Short English statement

Consider all quasivarieties contained in the class of torsion-free groups of nilpotency class at most $2$. Ordered by inclusion, they form a lattice whose meet is intersection and whose join is the quasivariety generated by the two classes. Budkin asks whether this lattice is distributive.

Sources

• Local UnsolvedMath registry, permanent ID $700932$, exact source problem code catalogue entry, title "Kourovka Notebook Problem 13.14." The supplied record labels the problem open; this label is supplied source metadata, not an independently verified mathematical or literature assessment.
• The Kourovka Notebook (Unsolved Problems in Group Theory), original problem $13.14$, proposed by A. I. Budkin in the 13th issue (1995). The supplied background also identifies the 21st issue edition (2026), arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors. These bibliographic details are reproduced from the record and have not been independently verified.