Kourovka Notebook Problem 19.15

Definitions and mathematical statement

For a class $M$ of groups, define \[ L(M)=\left\{G:\text{for every }a\in G,\ \langle a^G\rangle\in M\right\}, \qquad \langle a^G\rangle=\langle g^{-1}ag:g\in G\rangle. \] Here $\langle a^G\rangle$ is the normal closure of $a$ in $G$; the source's phrase "contained in $M$" is understood as membership in the class $M$.

For a finite nilpotent group $A$, let $qA$ be the quasivariety generated by $A$: the class of groups satisfying every group quasi-identity valid in $A$. A group quasi-identity is a universally quantified implication \[ \forall x_1,\ldots,x_n\quad \left(\bigwedge_{i=1}^{m}u_i(\mathbf{x})=v_i(\mathbf{x})\right) \Longrightarrow u(\mathbf{x})=v(\mathbf{x}), \] where the terms are group terms and the conjunction is finite, possibly empty.

For a group $H$, put $\gamma_1(H)=H$ and $\gamma_{k+1}(H)=[\gamma_k(H),H]$, where $[U,H]$ is generated by the commutators $u^{-1}h^{-1}uh$ with $u\in U$ and $h\in H$. The group $H$ is nilpotent if $\gamma_{c+1}(H)=\{1\}$ for some integer $c\geq 0$.

Question (A. I. Budkin, Kourovka Notebook 19.15). Does there exist a finite nilpotent group $A$ and a group $G$ such that \[ \left(\forall a\in G,\ \langle a^G\rangle\in qA\right) \quad\text{and}\quad \left(\forall c\geq 0,\ \gamma_{c+1}(G)\neq\{1\}\right)? \] Equivalently, can $L(qA)$ contain a non-nilpotent group for some finite nilpotent $A$? No finiteness or finite-generation assumption on $G$ is supplied.

Short English statement

Can a non-nilpotent group have every normal closure of a single element in the quasivariety generated by one finite nilpotent group? The finite group may be chosen, but it must be the same for all elements.

Sources

• Local UnsolvedMath registry, permanent ID $701563$, source code catalogue entry, title Kourovka Notebook Problem 19.15; supplied upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• Supplied attribution: A. I. Budkin, proposed in 2018; The Kourovka Notebook (Unsolved Problems in Group Theory), problem 19.15, 19th issue (2018), also identified in the supplied record with the 21st issue edition (2026), arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors.
• The registry labels the problem "open". This is a supplied status label, not an independently verified mathematical or literature assessment. The cited editions and attribution have not been independently checked.