Kourovka Notebook Problem 3.45

Definitions and mathematical statement

A group $G$ is torsion-free if, for every $g\in G$ and every integer $n\geq 1$, the equality $g^n=1$ implies $g=1$. It is locally nilpotent if every finitely generated subgroup is nilpotent. Here a group $H$ is nilpotent if its lower central series, defined by $\gamma_1(H)=H$ and $\gamma_{k+1}(H)=[\gamma_k(H),H]$, reaches the trivial subgroup after finitely many steps; $[A,B]$ denotes the subgroup generated by the commutators $a^{-1}b^{-1}ab$ with $a\in A$ and $b\in B$.

A subgroup $K\leq G$ is isolated in $G$ if \[ \forall g\in G\ \forall n\in{\mathbb{Z}}_{\geq 1},\qquad g^n\in K\ \Longrightarrow\ g\in K. \] Let $\mathfrak{X}$ be a hereditary radical of groups, and let $\mathfrak{X}(G)$ denote its associated radical subgroup of $G$. The supplied record does not specify the axioms for a hereditary radical or the construction of $\mathfrak{X}(G)$. These terms are therefore retained in the sense intended by the cited source; the quantification over radicals below requires that convention to be fixed before this is a fully self-contained formulation.

Question (B. I. Plotkin, 1969). For every hereditary radical $\mathfrak{X}$ and every locally nilpotent torsion-free group $G$, is $\mathfrak{X}(G)$ isolated in $G$? Equivalently, under the intended radical convention, must \[ \forall g\in G\ \forall n\in{\mathbb{Z}}_{\geq 1},\qquad g^n\in\mathfrak{X}(G)\ \Longrightarrow\ g\in\mathfrak{X}(G) \] hold?

Short English statement

Plotkin asks whether every hereditary radical subgroup of a locally nilpotent torsion-free group contains every element of the ambient group whose positive power it contains. The source labels the problem open; that label is supplied metadata, not an independently verified assessment. The intended definition of hereditary radical requires source clarification.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $700194$, title Kourovka Notebook Problem 3.45, exact source problem code catalogue entry; upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82. The record was created and updated on 2026-10-16T14:00:00Z and supplies the status open.
• The record attributes the problem to B. I. Plotkin, proposed in 1969, and cites The Kourovka Notebook (Unsolved Problems in Group Theory), 3rd issue (1969), problem 3.45, as included in the 21st-issue edition (2026), arXiv:1401.0300v48.
• The supplied collection description identifies E. I. Khukhro and V. D. Mazurov as editors of the Kourovka Notebook collection covering Issues 1--21 (1965--2026). The bibliographic attribution and edition information above are supplied source data and have not been independently verified.