Kourovka Notebook Problem 13.42

Definitions and mathematical statement

For a group $H$, define its lower central series by \[ \gamma_1(H)=H,\qquad \gamma_{n+1}(H)=[\gamma_n(H),H]\quad(n\geq 1), \] where $[K,H]$ is the subgroup generated by the commutators $k^{-1}h^{-1}kh$ with $k\in K$ and $h\in H$. The group $H$ is nilpotent if $\gamma_{c+1}(H)=\{1\}$ for some integer $c\geq 1$. For a set $X$, write $F(X)$ for the free group on $X$; a free nilpotent group has the form \[ N=F(X)/\gamma_{c+1}(F(X)),\qquad c\geq 1. \] The source does not specify restrictions on the rank $\lvert X\rvert$ or on $c$.

Write $T_A(N)$ provisionally for the tensor $A$-completion referred to in A. G. Myasnikov and V. N. Remeslennikov, Siberian Math. J. 35, no. 5 (1994), 986--996. The supplied record does not define this construction, the nature of $A$, or its admissibility hypotheses; this notation is a reference to that missing definition and does not specify a construction independently.

The requested assertion is that a free nilpotent group can have a non-nilpotent tensor $A$-completion. With admissibility understood to include all hypotheses of the cited construction, its provisional existential formulation is \[ \exists\,(A,X,c)\ \text{admissible},\quad c\geq 1,\qquad \forall n\geq 1,\quad \gamma_{n+1}\!\left(T_A\!\left(F(X)/\gamma_{c+1}(F(X))\right)\right)\neq\{1\}. \] This formulation requires clarification of the admissible coefficient objects and input groups. In particular, the supplied wording does not determine whether $A$ may be chosen or is intended to be fixed subject to further hypotheses.

Short English statement

The problem asks for a free nilpotent group whose tensor $A$-completion is not nilpotent. A precise unconditional formulation requires the missing definition of that completion and the hypotheses on $A$. The supplied registry labels the problem open and records an explicitly unverified AI solution claim; neither assessment is independently verified here.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $700960$, exact source problem code catalogue entry, title Kourovka Notebook Problem 13.42; upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• The Kourovka Notebook (Unsolved Problems in Group Theory), original problem 13.42, 13th issue (1995); attributed to A. G. Myasnikov and V. N. Remeslennikov. The record cites the 21st issue edition (2026), arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors.
• A. G. Myasnikov and V. N. Remeslennikov, Siberian Math. J. 35, no. 5 (1994), 986--996, cited by the source for tensor $A$-completion. No article title or definition is supplied.
• The supplied discussion records: [Unverified AI claim] an example is produced in M. Kinyon and J. Urban, Preprint (2026), https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2026/09/kourovka-mathematics.pdf. This is explicitly an unverified AI solution claim, not an established solution in this draft.
• The supplied status review is dated 2026-10-16, gives canonical status open and evidence unrefereed_claim, and reports a $\#$ marking for an unverified AI solution claim in the October 2026 21st issue. This assessment is supplied dated curation, not independently verified.