Kourovka Notebook Problem 20.18

Definitions and mathematical statement

For a prime $p$ and an integer $k\geq 2$, let \[ R_{p^k}=\{G: G\models \forall x\,x^{p^k}=1\text{ and }\forall x,y,z\,[[x,y],z]=1\}, \] where $[x,y]=x^{-1}y^{-1}xy$. Thus $R_{p^k}$ is the variety of groups of nilpotency class at most $2$ and exponent dividing $p^k$. These bounds are the conventional interpretation of the source's phrase "variety of class $2$ nilpotent groups of exponent $p^k$." Write $G' = \langle [x,y]:x,y\in G\rangle$ for the derived subgroup. For a finite group $H$, $\exp(H)$ is the least positive integer $e$ such that $h^e=1$ for every $h\in H$.

A group quasi-identity is a universally quantified implication \[ \forall x_1,\ldots,x_n\;\left(\bigwedge_{j=1}^{m}u_j=v_j\ \Longrightarrow\ u=v\right), \] where all displayed expressions are group terms and the conjunction may be empty. A quasivariety is a class of groups axiomatized by quasi-identities. Let $\mathsf Q(F)$ denote the smallest quasivariety containing a group $F$, equivalently the class of groups satisfying every quasi-identity valid in $F$.

A basis of quasi-identities for a quasivariety $\mathcal K$ is a set $B$ whose group models are exactly $\mathcal K$. It is independent if, for every $\varphi\in B$, there exists a group satisfying $B\setminus\{\varphi\}$ and failing $\varphi$. This defines independence over the group axioms. A relative basis within $R_{p^k}$ instead has $\mathcal K=\{G\in R_{p^k}:G\models B\}$, and relative independence requires each witnessing group to belong to $R_{p^k}$. The supplied record does not specify which meaning of basis and independence is intended; that qualification remains unresolved.

Question (A. I. Budkin, 2022). Is it true that, for every prime $p$ and every integer $k\geq 2$, there exist infinitely many pairwise distinct subquasivarieties $\mathcal K_i\subseteq R_{p^k}$, indexed by $i\geq 1$, and finite groups $F_i\in R_{p^k}$ such that \[ \mathcal K_i=\mathsf Q(F_i),\qquad \exp(F_i')=p^k, \] and each $\mathcal K_i$ has no independent basis of quasi-identities? The basis convention must be resolved as described above. The equality $\exp(F_i')=p^k$ retains the source's stated exponent requirement as an exact exponent. The universal quantifier includes $p=2$; the source supplies no exception for that prime.

Short English statement

For every prime $p$ and integer $k\geq 2$, Budkin asks whether there are infinitely many distinct quasivarieties within $R_{p^k}$, each generated by a finite group whose derived subgroup has exponent exactly $p^k$, and each lacking an independent basis of quasi-identities. The source leaves unresolved whether the basis and independence are meant over all groups or relative to $R_{p^k}$.

Sources

• Local UnsolvedMath registry, permanent ID $701676$, exact source problem code catalogue entry, title Kourovka Notebook Problem 20.18; supplied upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82. The registry labels the problem open; this is a supplied status, not an independently verified mathematical assessment.
• The Kourovka Notebook (Unsolved Problems in Group Theory), Problem 20.18, proposed by A. I. Budkin in 2022. The supplied background identifies the 20th issue (2022) and 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). These bibliographic details are reproduced from the record without independent source consultation.