Kourovka Notebook Problem 15.74

Definitions and mathematical statement

For a group $G$, define its lower central series by $\gamma_1(G)=G$ and $\gamma_{i+1}(G)=[\gamma_i(G),G]$, where $[H,G]$ is generated by the commutators $h^{-1}g^{-1}hg$ with $h\in H$ and $g\in G$. The group $G$ has nilpotence class exactly $p$ if $\gamma_{p+1}(G)=\{1\}$ and $\gamma_p(G)\ne\{1\}$.

Let $\operatorname{Norm}(G)$ be the lattice of normal subgroups of $G$, with meet $N\cap M$ and join $NM$. For an abelian group $A$, let $\operatorname{Sub}(A)$ be its subgroup lattice, with meet $B\cap C$ and join $BC$. Here a lattice embedding means an injective map preserving binary meets and joins. The supplied source does not explicitly specify whether preservation of the least and greatest elements is also required; no such additional requirement is imposed in this formulation.

Problem. For every prime $p$, find a finite group $G$ of order $p^n$ for some integer $n\geq 1$, of nilpotence class exactly $p$, such that, for every abelian group $A$, there is no injective map \[ \phi:\operatorname{Norm}(G)\longrightarrow\operatorname{Sub}(A) \] satisfying, for all $N,M\trianglelefteq G$, \[ \phi(N\cap M)=\phi(N)\cap\phi(M),\qquad \phi(NM)=\phi(N)\phi(M). \] The abelian group $A$ is not assumed finite.

The supplied statement says that the problem is solved for $p=2$ and $p=3$. This is a source-reported qualification, not an independently verified result; the registry labels the overall problem as open.

Short English statement

For each prime $p$, find a finite $p$-group of nilpotence class exactly $p$ whose normal-subgroup lattice cannot embed, preserving meets and joins, into the subgroup lattice of any abelian group, including an infinite one. The supplied source reports solutions for $p=2,3$ and labels the overall problem open.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $701162$, exact source code catalogue entry, title Kourovka Notebook Problem 15.74. The supplied registry status is open.
• The Kourovka Notebook (Unsolved Problems in Group Theory), original problem 15.74, 15th issue (2002); proposed by P. P. Pálfy. The supplied background also identifies the 21st issue edition (2026), arXiv:1401.0300v48. The collection description attributes editing to E. I. Khukhro and V. D. Mazurov.
• The parenthetical report that the cases $p=2,3$ are solved comes from the supplied statement. No separate references for these cases are supplied. The cited source and its reported status have not been independently checked for this draft.