Kourovka Notebook Problem 11.73

Definitions and mathematical statement

A variety of groups $\mathcal V$ is the class of all groups satisfying some specified collection of group identities. A group $G\in\mathcal V$ is relatively free in $\mathcal V$ if there are a set $X$ and a map $\iota:X\to G$ whose image generates $G$, such that for every $H\in\mathcal V$ and every map $f:X\to H$, there is a unique homomorphism $\varphi:G\to H$ with $\varphi\circ\iota=f$.

A group is finitely presented if, as an ordinary group, it admits a presentation $\langle S\mid R\rangle$ with both $S$ and $R$ finite. A group $N$ is nilpotent if its lower central series, defined by $\gamma_1(N)=N$ and $\gamma_{j+1}(N)=[\gamma_j(N),N]$, satisfies $\gamma_{c+1}(N)=\{1\}$ for some integer $c\geq 0$. Here $[A,B]$ denotes the subgroup generated by the commutators $a^{-1}b^{-1}ab$, with $a\in A$ and $b\in B$. A group is virtually nilpotent if it has a nilpotent subgroup of finite index.

Question (A. Yu. Olshanskii, 1990). For every variety of groups $\mathcal V$ and every group $G$ relatively free in $\mathcal V$, does finite presentability of $G$ imply that there exist a subgroup $N\leq G$ and an integer $c\geq 0$ such that \[ [G:N]<\infty \qquad\text{and}\qquad \gamma_{c+1}(N)=\{1\}? \] The terms "relatively free" and "finitely presented" are interpreted here in their standard senses defined above; the supplied record gives no additional restrictions on the variety or the free generating set.

Short English statement

Must every finitely presented relatively free group contain a nilpotent subgroup of finite index? The question is attributed to A. Yu. Olshanskii (1990). The supplied registry labels it open; that status has not been independently verified.

Sources

• Local UnsolvedMath registry, permanent ID $700765$, source problem code catalogue entry, title Kourovka Notebook Problem 11.73; supplied upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), Problem 11.73; proposed by A. Yu. Olshanskii. 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 of the Kourovka Notebook collection covering Issues 1--21 (1965--2026). The bibliographic information and open-status label are supplied source data, not an independently verified literature assessment.