Kourovka Notebook Problem 2.6

Definitions and mathematical statement

Let $G$ be a group. Write $\mathcal C_{\mathrm{gen}}(G)$ for the set called the generalized central elements of $G$ in the supplied source, and write $\mathcal N_{\mathrm{nil}}(G)$ for its nil-kernel. These symbols are placeholders for the source terminology: the supplied record defines neither of these notions nor the term $NR$-group. Consequently, the formulation below requires those definitions before it can be a self-contained mathematical statement.

The supplied record states the forward implication \[ G\text{ is an }NR\text{-group} \quad\Longrightarrow\quad \mathcal C_{\mathrm{gen}}(G)=\mathcal N_{\mathrm{nil}}(G). \] The question, attributed to V. G. Vilyatser, asks whether the converse holds: for every group $G$, does \[ \mathcal C_{\mathrm{gen}}(G)=\mathcal N_{\mathrm{nil}}(G) \quad\Longrightarrow\quad G\text{ is an }NR\text{-group}? \] The record specifies no restriction to finite groups or to another class of groups; any restrictions implicit in the undefined terminology remain unresolved.

Short English statement

V. G. Vilyatser asks whether equality of a group's generalized central elements and its nil-kernel forces the group to be an $NR$-group. The supplied record states the reverse implication. Definitions of all three specialized notions are missing, so this draft retains the question provisionally and requires mathematical review.

Sources

• Local UnsolvedMath registry, supplied record with permanent ID $700085$, exact source problem code catalogue entry, title Kourovka Notebook Problem 2.6; upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• Supplied bibliographic attribution: The Kourovka Notebook (Unsolved Problems in Group Theory), problem 2.6, second issue (1967), proposed by V. G. Vilyatser; cited in the record through the twenty-first issue edition (2026), arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors.
• The local record labels the problem open. This is supplied registry status, not an independently verified literature assessment. The cited source and its terminology have not been independently checked for this draft.