Kourovka Notebook Problem 2.5

Definitions and mathematical statement

Let $G$ be a group, with no finiteness assumption. A group is locally nilpotent if each of its finitely generated subgroups is nilpotent. Write $R_{\mathrm{ln}}(G)$ for the locally nilpotent radical of $G$, namely its largest normal locally nilpotent subgroup.

The supplied record uses Plotkin's terms "nil-element," "nil-automorphism," and "locally stable automorphism" without defining them. To retain these unresolved source-specific meanings, write \[ \begin{aligned} \mathcal N(G)&=\{g\in G:g\text{ is a nil-element in the source's sense}\},\\ \mathcal A_{\mathrm{nil}}(G)&=\{\alpha\in\operatorname{Aut}(G):\alpha\text{ is a nil-automorphism in the source's sense}\},\\ \mathcal A_{\mathrm{ls}}(G)&=\{\alpha\in\operatorname{Aut}(G):\alpha\text{ is locally stable in the source's sense}\}. \end{aligned} \] These are placeholders for the source terminology, not independent definitions of those three properties.

According to the supplied statement, $G$ is an $NR$-group precisely when \[ \mathcal N(G)=R_{\mathrm{ln}}(G). \] The source also asserts that this condition is equivalent to \[ \operatorname{Inn}(G)\subseteq\mathcal A_{\mathrm{ls}}(G), \] where $\operatorname{Inn}(G)$ is the group of inner automorphisms of $G$.

Question (V. G. Vilyatser, 1967). Does there exist a group $G$ and an automorphism $\alpha\in\operatorname{Aut}(G)$ such that \[ \mathcal N(G)=R_{\mathrm{ln}}(G) \quad\text{and}\quad \alpha\in\mathcal A_{\mathrm{nil}}(G)\setminus\mathcal A_{\mathrm{ls}}(G)? \] A self-contained mathematical formulation requires the missing definitions of the three source-specific properties. The displayed question preserves the logical form of the supplied problem without selecting unconfirmed interpretations.

Short English statement

Vilyatser asks whether an $NR$-group can admit a nil-automorphism that is not locally stable. The source characterizes $NR$-groups by equality of their nil-elements with their locally nilpotent radical, and equivalently by local stability of every inner automorphism. The meanings of nil-element, nil-automorphism, and local stability require clarification from the source.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $700084$, exact source problem code catalogue entry, title "Kourovka Notebook Problem 2.5." Upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• Source attribution supplied by the registry: The Kourovka Notebook (Unsolved Problems in Group Theory), 2nd issue (1967), problem 2.5; proposed by V. G. Vilyatser. The definition of an $NR$-group is attributed to Plotkin.
• Edition identified in the supplied background: 21st issue edition (2026), arXiv:1401.0300v48. The collection description names E. I. Khukhro and V. D. Mazurov as editors. This bibliographic attribution is supplied source data and has not been independently verified.
• The registry labels the problem "open" and records an update timestamp of 2026-10-16T14:00:00Z. This is supplied registry metadata, not an independently verified literature assessment or a stated literature-check date.