Kourovka Notebook Problem 17.31

Definitions and mathematical statement

Let $G$ be a group with identity $1$. Its derived subgroup is \[ G'=[G,G]=\langle x^{-1}y^{-1}xy:x,y\in G\rangle. \] Define $G^{(0)}=G$ and $G^{(n+1)}=[G^{(n)},G^{(n)}]$ for integers $n\geq 0$. The group $G$ is soluble if $G^{(n)}=\{1\}$ for some integer $n\geq 0$. It is right-orderable if there exists a strict total order $<$ on $G$ such that, for every $x,y,z\in G$, \[ x<y\quad\Longrightarrow\quad xz<yz. \] The quotient $G/G'$ is its abelianization.

The supplied question asks whether there exists a soluble right-orderable group $G$ such that \[ |G/G'|=[G:G']<\infty. \] No bound on the derived length or on this finite index is specified. Neither finite generation nor an order preserved by left multiplication is required in the supplied wording.

Unresolved qualification. The supplied statement does not require $G\neq\{1\}$, so its literal quantification includes the trivial group. Whether a nontrivial group is intended must be clarified before this can be treated as a precise formulation of the intended open problem. No nontriviality or infinitude hypothesis has been inserted here.

Short English statement

The question asks whether a soluble group admitting a right-invariant total order can have finite abelianization. The source does not specify that the group must be nontrivial; this omission leaves the intended scope unresolved.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $701333$, exact source code catalogue entry, title Kourovka Notebook Problem 17.31; upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• Attributed in the supplied record to V. V. Bludov and A. H. Rhemtulla, proposed in 2010.
• Supplied reference: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), original problem 17.31; also identified with 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 supplied attribution and have not been independently verified.
• The local record labels the problem open and gives an update timestamp of 2026-10-16T14:00:00Z. This is supplied registry metadata, not an independently verified literature assessment. No separate literature-verification date or external reference URL is supplied.