Kourovka Notebook Problem 17.91

Definitions and mathematical statement

For a group $X$, define its derived series by $X^{(0)}=X$ and $X^{(i+1)}=[X^{(i)},X^{(i)}]$, where $[H,H]$ is the subgroup generated by all commutators of elements of $H$. A group $X$ is soluble if $X^{(n)}=\{1\}$ for some integer $n\geq 0$. Its derived length is \[ d(X)=\min\{n\in{\mathbb{Z}}_{\geq 0}:X^{(n)}=\{1\}\}. \] In particular, $d(\{1\})=0$. A maximal subgroup $M$ of $G$ means a proper subgroup with no subgroup strictly between $M$ and $G$.

1. (a) Does there exist an absolute constant $k\in{\mathbb{Z}}_{\geq 0}$ such that, for every finite soluble group $G$ and every maximal subgroup $M<G$, \[ d(G)-d(M)\leq k? \]
2. (b) If such a constant exists, determine the least possible $k$.

Here $k$ must be independent of both $G$ and $M$. Restricting $k$ to nonnegative integers does not change the boundedness question, since the differences are nonnegative integers. The supplied record labels the problem open; this is a supplied status, not an independently verified literature assessment.

Short English statement

Is the difference between the derived length of a finite soluble group and that of any maximal subgroup bounded by a universal constant? If so, determine the smallest such constant.

Sources

• Local UnsolvedMath registry, permanent ID $701393$, exact source code catalogue entry, title Kourovka Notebook Problem 17.91. The supplied record labels the problem open and records an update date of 2026-10-16.
• Attributed in the supplied record to V. S. Monakhov, proposed in 2010: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), Problem 17.91. The background also identifies the 21st issue edition (2026), arXiv:1401.0300v48. The collection description names E. I. Khukhro and V. D. Mazurov as editors. These bibliographic details are supplied attribution and have not been independently verified.