Kourovka Notebook Problem 11.115

Definitions and mathematical statement

Let $X$ be a noncyclic free group, with no restriction in the supplied statement that its rank be finite. Let $1\ne N\trianglelefteq X$ and let $N\le T<X$. For subgroups $A,B\le X$, define \[ [A,B]=\langle a^{-1}b^{-1}ab\mid a\in A,\ b\in B\rangle. \] Here $<$ denotes proper subgroup inclusion. In particular, $[T,N]\le [X,N]$.

The source asks whether, for every such triple $(X,N,T)$, the additional hypothesis that $N$ is "not maximal in $X$" implies \[ [T,N]<[X,N]. \] Under the interpretation that maximality refers to all subgroups of $X$, the question is precisely whether \[ \left(\exists H\le X\;\;N<H<X\right) \quad\Longrightarrow\quad [T,N]\ne[X,N] \] holds for every noncyclic free group $X$, every nontrivial normal subgroup $N$ of $X$, and every subgroup $T$ satisfying $N\le T<X$.

Unresolved qualification. The supplied record does not specify whether "maximal" instead means maximal among proper normal subgroups. Under that interpretation, the additional hypothesis would be \[ \exists H\trianglelefteq X\quad N<H<X. \] These two hypotheses need not coincide; the intended interpretation requires source review. The source permits $T=N$ and does not require $T$ to be normal in $X$.

Short English statement

V. P. Shaptala asks whether a nontrivial normal subgroup $N$ of a noncyclic free group $X$, subject to the source's nonmaximality condition, satisfies $[T,N]<[X,N]$ for every proper subgroup $T$ containing $N$. The meaning of nonmaximality requires clarification. The supplied registry retains an open label and records an explicitly unverified AI claim of a negative answer; that claim is not treated here as an established result.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $700807$, exact source code catalogue entry, title Kourovka Notebook Problem 11.115. Attribution: V. P. Shaptala, proposed in 1990.
• Supplied primary-source attribution: The Kourovka Notebook (Unsolved Problems in Group Theory), 11th issue (1990), Problem 11.115; also identified in the supplied record with the 21st issue edition (2026), arXiv:1401.0300v48. The collection description names E. I. Khukhro and V. D. Mazurov as editors.
• Supplied discussion and status-review note: the 21st issue (October 2026) marks the problem with $\#$ and reports an explicitly unverified AI claim that the answer is negative, attributed to M. Kinyon and J. Urban, Preprint, 2026: https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2026/09/kourovka-mathematics.pdf. This is supplied attribution of an unverified claim, not verification of a counterexample or solution.
• The supplied status review is dated 2026-10-16, retains canonical status open, and classifies its evidence as unrefereed_claim. This assessment is supplied dated curation, not independently verified literature assessment. No cited source was independently consulted for this draft.