Kourovka Notebook Problem 18.28

Definitions and mathematical statement

The source does not specify the ambient class of groups or define a formation. The following makes its question precise relative to an ambient class $\mathcal U$, using closure under finite subdirect products. Whether this is the intended convention requires review; in particular, the source does not explicitly restrict to finite groups.

Fix a nonempty class $\mathcal U$ of groups closed under isomorphisms, homomorphic images, and finite subdirect products. A formation in $\mathcal U$ is a nonempty subclass $\mathcal F\subseteq\mathcal U$ closed under these operations. Here a subdirect product of $G_1,\ldots,G_n$, with $n\geq 1$, is a subgroup $S\leq G_1\times\cdots\times G_n$ whose projection onto each factor is surjective. For $G\in\mathcal U$, define \[ \operatorname{form}_{\mathcal U}(G) =\bigcap\{\mathcal F:\mathcal F\text{ is a formation in }\mathcal U,\ G\in\mathcal F\}. \] A formation is one-generator if it equals $\operatorname{form}_{\mathcal U}(G)$ for some group $G\in\mathcal U$. This refers to generation of a formation by one group and does not require that group to be cyclic. A subformation of $\mathcal F$ is a formation $\mathcal H$ with $\mathcal H\subseteq\mathcal F$.

Question (V. P. Burichenko, Kourovka Notebook 18.28). For the intended ambient class and formation convention, is it true that \[ \forall G\in\mathcal U\quad \forall\mathcal H\subseteq\operatorname{form}_{\mathcal U}(G), \qquad \mathcal H\text{ is a formation} \ \Longrightarrow\ \exists H\in\mathcal U\quad \mathcal H=\operatorname{form}_{\mathcal U}(H)? \] The choice of $H$ may depend on both $G$ and $\mathcal H$. This is a formulation schema pending confirmation of the source's ambient class and closure convention, not an assertion that the question ranges over every possible $\mathcal U$.

Short English statement

Must every subformation of a formation generated by a single group itself be generated by a single group? The source leaves the ambient class of groups and the formation convention unstated.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $701457$, exact source problem code catalogue entry, title Kourovka Notebook Problem 18.28; upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• The supplied record attributes the problem to V. P. Burichenko, proposed in 2014, and cites The Kourovka Notebook (Unsolved Problems in Group Theory), 18th issue (2014), problem 18.28, with the 21st issue edition (2026), arXiv:1401.0300v48. Its collection description names E. I. Khukhro and V. D. Mazurov as editors.
• The registry labels the problem open. This is supplied source metadata, not an independently verified assessment. The cited edition and its surrounding definitions have not been independently consulted for this draft.