Kourovka Notebook Problem 20.105

Definitions and mathematical statement

Let $p$ be a prime. A group $G$ is a $p$-group if every element has finite order equal to a power of $p$. Its commutator subgroup is \[ [G,G]=\langle x^{-1}y^{-1}xy:x,y\in G\rangle, \] and $G$ is perfect if $G=[G,G]$.

For any group $H$, define its transfinite upper central series by \[ Z_0(H)=\{1\},\qquad Z_{\alpha+1}(H)/Z_\alpha(H)=Z\left(H/Z_\alpha(H)\right),\qquad Z_\lambda(H)=\bigcup_{\alpha<\lambda}Z_\alpha(H) \] for every ordinal $\alpha$ and every nonzero limit ordinal $\lambda$, where $Z(K)$ denotes the center of a group $K$. The group $H$ is nilpotent if $Z_n(H)=H$ for some nonnegative integer $n$, and hypercentral if $Z_\alpha(H)=H$ for some ordinal $\alpha$. A group is locally nilpotent if each of its finitely generated subgroups is nilpotent.

The supplied question asks whether there exist a prime $p$ and a group $G$ such that \[ \begin{gathered} G\text{ is a perfect, locally nilpotent }p\text{-group},\\ \forall H<G\;\exists\text{ an ordinal }\alpha_H\quad Z_{\alpha_H}(H)=H. \end{gathered} \] Here $H<G$ means any proper subgroup, without a normality or finite-generation restriction, and $\alpha_H$ may depend on $H$.

The source does not explicitly require $G\ne\{1\}$. Under the definitions above, the trivial group is permitted by the stated conditions. Whether a nontrivial group is intended is therefore an unresolved qualification; the additional hypothesis $G\ne\{1\}$ has not been silently inserted.

Short English statement

Does there exist, for some prime $p$, a perfect locally nilpotent $p$-group all of whose proper subgroups are hypercentral? The source leaves nontriviality unstated, so the intended requirement concerning $G\ne\{1\}$ needs clarification.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $701763$, exact source code catalogue entry, titled Kourovka Notebook Problem 20.105. The registry labels the problem open; this is supplied source metadata, not an independently verified assessment.
• The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), Problem 20.105; the supplied background also identifies the 21st issue edition (2026), arXiv:1401.0300v48. The collection description names E. I. Khukhro and V. D. Mazurov as editors. These bibliographic details are supplied and have not been independently checked.
• Proposed by N. Trabelsi; proposed year 2022, according to the supplied record.