Kourovka Notebook Problem 3.43

Definitions and mathematical statement

Let $\mu$ be an infinite cardinal, identified with its initial ordinal. To make the source's terminology explicit, adopt the usual continuous transfinite-series convention: a group $H$ is $\mu$-overnilpotent if, for every $h\in H$, there exist an ordinal $\lambda<\mu$ and subgroups $(H_\alpha)_{\alpha\leq\lambda}$ satisfying \[ H_0=\langle h\rangle,\qquad H_\lambda=H,\qquad H_\alpha\trianglelefteq H_{\alpha+1}\quad(\alpha<\lambda), \] with the family ascending and \[ H_\delta=\bigcup_{\alpha<\delta}H_\alpha \quad\text{for every nonzero limit ordinal }\delta\leq\lambda. \] Thus each cyclic subgroup begins a possibly transfinite normal series reaching $H$. Normality here means normality in the next subgroup. The source does not explicitly specify continuity or its convention for measuring series length; these conventions require confirmation.

Write $\mathcal R_\mu$ for the class of $\mu$-overnilpotent groups. The source states that $\mathcal R_\mu$ is a radical class. Under the standard radical-class interpretation, call $G$ $\mu$-semisimple if \[ \forall N\trianglelefteq G,\qquad N\in\mathcal R_\mu\ \Longrightarrow\ N=\{1\}. \] This interpretation of semisimplicity is not explicitly defined in the supplied record.

The question attributed to B. I. Plotkin (1969) is whether \[ \forall\text{ infinite cardinals }\mu_1<\mu_2\quad \exists\text{ a group }G:\qquad G\in\mathcal R_{\mu_2} \quad\text{and}\quad \forall N\trianglelefteq G\, \left(N\in\mathcal R_{\mu_1}\Longrightarrow N=\{1\}\right). \] The supplied wording does not explicitly require $G\neq\{1\}$. Whether nontriviality is intended must be resolved before treating this as a precise separation problem; it has not been silently added to the displayed question.

The source's Remark of 2001 attributes to S. Vovsi (1972) the weaker assertion \[ \forall\text{ infinite cardinals }\mu_1<\mu_2\quad \exists G\in\mathcal R_{\mu_2}\setminus\mathcal R_{\mu_1}. \] This supplied historical assertion is distinct from the requested semisimplicity condition. The record also reports an affirmative answer as an explicitly unverified AI solution claim; it does not supply a verified resolution.

Short English statement

For every pair of infinite cardinals $\mu_1<\mu_2$, Plotkin asks for a $\mu_2$-overnilpotent group having no nontrivial normal $\mu_1$-overnilpotent subgroup. The intended nontriviality of the group and the precise transfinite-series conventions need confirmation. The supplied 2001 remark reports a weaker result, and the reported 2026 affirmative claim is explicitly unverified.

Sources

• Local UnsolvedMath registry: permanent ID $700192$, exact source problem code catalogue entry, title Kourovka Notebook Problem 3.43; supplied complete source record, upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• B. I. Plotkin, Problem 3.43, The Kourovka Notebook (Unsolved Problems in Group Theory), 3rd issue (1969). The supplied record identifies the collection as Issues 1--21, edited by E. I. Khukhro and V. D. Mazurov, and cites the 21st issue edition (2026), arXiv:1401.0300v48.
• S. Vovsi, Sov. Math. Dokl. 13 (1972), 408--410. The supplied Remark of 2001 attributes the weaker strict-inclusion result to this reference; no article title is supplied.
• Achyuth Jayadevan, Preprint (2026), https://zenodo.org/records/22721386. The supplied background and status-review notes describe this as an unverified AI solution claiming an affirmative answer, marked with $\#$ in the October 2026 21st issue; no preprint title is supplied.
• Supplied status curation dated 2026-10-16: the registry status and canonical status are open, with evidence classified as unrefereed_claim. This is supplied dated curation, not an independently verified literature assessment. None of the cited sources has been independently checked for this draft.