Kourovka Notebook Problem 16.18

Definitions and mathematical statement

For a group $G$, define its derived series by $G^{(0)}=G$ and $G^{(i+1)}=[G^{(i)},G^{(i)}]$, where $[H,H]$ is the subgroup generated by the commutators of elements of $H$. The group $G$ is soluble of derived length exactly $n$ if \[ G^{(n)}=\{1\}\qquad\text{and}\qquad G^{(n-1)}\ne\{1\}. \] A total order $<$ on $G$ is left invariant if $x<y$ implies $gx<gy$ for all $g,x,y\in G$; it is right invariant if $x<y$ implies $xg<yg$. It is invariant on both sides if it has both properties. The supplied record does not specify which invariance convention is intended by "linearly orderable."

Fix an intended order-invariance convention, and let $\mathcal O(G)$ denote the total orders on $G$ satisfying that convention. Under the usual existential interpretation, a subgroup $H\le G$ is relatively convex if there exists $<\in\mathcal O(G)$ such that \[ \forall h_1,h_2\in H\ \forall g\in G,\qquad h_1<g<h_2\ \Longrightarrow\ g\in H. \] The witnessing order may depend on $H$. Define \[ \mathcal R(G)=\{H\le G: H\ne G,\ H\trianglelefteq G,\ H\text{ is relatively convex}\}. \] With "proper" interpreted literally as $H\ne G$, the requested existence assertions are \[ \begin{aligned} \text{(a)}\quad &\exists G:\quad \mathcal O(G)\ne\varnothing,\quad G^{(3)}=\{1\},\quad G^{(2)}\ne\{1\},\quad |\mathcal R(G)|=1;\\ \text{(b)}\quad &\forall n\in\mathbb Z,\ n>4,\ \exists G:\quad \mathcal O(G)\ne\varnothing,\quad G^{(n)}=\{1\},\quad G^{(n-1)}\ne\{1\},\quad |\mathcal R(G)|=1. \end{aligned} \] Here the group in part (b) may depend on $n$.

There is an unresolved convention in the uniqueness requirement: the trivial subgroup is proper whenever $G\ne\{1\}$, and it is convex for every total order. If the source intends "proper" to mean nontrivial and proper, then the condition instead becomes \[ \left|\{H\in\mathcal R(G):H\ne\{1\}\}\right|=1. \] The supplied record does not resolve this distinction or explicitly define relative convexity, so the displayed assertions are conditional formulations rather than a uniquely determined interpretation of the source.

The source additionally states that groups with the intended properties exist for $n=2$ and $n=4$. This is retained as a supplied background assertion, without independent verification.

Short English statement

The problem asks whether a linearly orderable soluble group with exactly one proper normal relatively convex subgroup can have derived length $3$, and whether such a group exists for every integer derived length greater than $4$. The source reports examples of lengths $2$ and $4$. The intended order-invariance convention and whether the unique subgroup must be nontrivial require clarification.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $701210$, exact source problem code catalogue entry, titled "Kourovka Notebook Problem 16.18." The record labels the problem open; this is supplied registry status, not an independently verified mathematical assessment.
• V. V. Bludov, proposer of Problem 16.18 in The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), as attributed in the supplied record.
• The supplied background also identifies the 21st issue edition (2026), arXiv:1401.0300v48. The collection description names E. I. Khukhro and V. D. Mazurov as editors. No source text beyond the supplied record was consulted.
• Registry provenance: upstream ID $701210$, upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82, and supplied record update timestamp 2026-10-16T14:00:00Z. This timestamp is registry metadata, not a literature-verification date.