Kourovka Notebook Problem 2.67

Definitions and mathematical statement

For a group $G$, let $L(G)$ denote the lattice of all subgroups of $G$, ordered by inclusion, with meet given by intersection and join given by the subgroup generated by the union. For a specified class $\mathcal H$ of groups, say that $G$ is determined by its subgroup lattice relative to $\mathcal H$ if \[ \forall H\in\mathcal H,\qquad L(H)\cong L(G)\ \Longrightarrow\ H\cong G, \] where the first isomorphism is a lattice isomorphism and the second is a group isomorphism. Taking $\mathcal H$ to be all groups gives the usual unrestricted interpretation; the supplied record does not specify whether a restricted comparison class is intended.

A group $G$ is nilpotent if its lower central series, defined by $\gamma_1(G)=G$ and $\gamma_{n+1}(G)=[\gamma_n(G),G]$, reaches the trivial subgroup after finitely many steps. It is torsion-free if, for every $g\in G$ and every positive integer $n$, the equality $g^n=1$ implies $g=1$.

The source asks for conditions ensuring that a nilpotent product $P$ of pure nilpotent groups, drawn from suitable classes, is determined by its subgroup lattice. Thus, after specifying the meaning of pure nilpotent group, the nilpotent-product construction, the permitted factor classes, and the comparison class $\mathcal H$, the requested conclusion is \[ \forall H\in\mathcal H,\qquad L(H)\cong L(P)\ \Longrightarrow\ H\cong P. \] This is a request to find sufficient conditions, not a single universally quantified conjecture with specified hypotheses. The supplied record does not define purity, specify the number of factors or a nilpotency-class parameter for the product, or identify the intended classes. These omissions remain unresolved.

The source additionally reports that the desired determination is known when the product $P$ is torsion-free. This is preserved as a supplied background assertion within the source's unspecified setting, rather than asserted here as a verified theorem for every possible interpretation.

Short English statement

Find sufficient conditions under which a nilpotent product of pure nilpotent groups is determined, up to group isomorphism, by its subgroup lattice. The source reports the torsion-free case as known. The meanings of purity, the product construction, the allowed factor classes, and the comparison class require clarification before a fully specified statement can be given.

Sources

• Local UnsolvedMath registry: permanent ID $700137$, exact source problem code catalogue entry, title Kourovka Notebook Problem 2.67; upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82. The registry labels the problem open; this is supplied classification, not an independently verified status assessment.
• Supplied source attribution: The Kourovka Notebook (Unsolved Problems in Group Theory), original problem 2.67, second issue (1967), as recorded in the twenty-first issue edition (2026), arXiv:1401.0300v48. The collection description names E. I. Khukhro and V. D. Mazurov as editors. No source consultation or independent verification is claimed.
• Proposed in 1967 by L. E. Sadovski, with an incomplete TeX suffix in both supplied proposer fields. The exact malformed spelling is retained in the accompanying ambiguity record; the intended final letters or diacritics have not been reconstructed.
• The torsion-free background assertion is supplied without a separate theorem reference, proof, or precise hypotheses. No external URL or separate literature-verification date is supplied.