Kourovka Notebook Problem 14.81

Definitions and mathematical statement

In the usual setting of formations of finite groups, a formation is a nonempty class $\mathcal F$ of finite groups, closed under isomorphisms, homomorphic images, and finite subdirect products. Here a subdirect product of $G_1,\ldots,G_r$, with $r\geq 1$, is a subgroup $D\leq G_1\times\cdots\times G_r$ whose projection onto every factor is surjective. For a finite group $G$, define its generated formation by \[ \operatorname{form}(G)=\bigcap\{\mathcal F: \mathcal F\text{ is a formation of finite groups and }G\in\mathcal F\}. \] A subformation of $\operatorname{form}(G)$ is a formation $\mathcal H$ with $\mathcal H\subseteq\operatorname{form}(G)$.

The supplied assertion is that, for every finite group $G$, the formation $\operatorname{form}(G)$ has only finitely many $S_n$-closed subformations. The source record does not define $S_n$-closed or explicitly specify the ambient universe for formations, so a completely unqualified formalization is unavailable from the supplied record alone.

If $S_n$-closed means closed under taking subnormal subgroups, the assertion has the following precise formulation. Write $L\mathrel{\trianglelefteq\trianglelefteq}K$ when there are an integer $r\geq 0$ and a chain \[ L=L_0\trianglelefteq L_1\trianglelefteq\cdots\trianglelefteq L_r=K. \] Call a formation $\mathcal H$ subnormal-subgroup-closed when \[ \forall K\in\mathcal H\;\forall L\leq K,\qquad L\mathrel{\trianglelefteq\trianglelefteq}K\ \Longrightarrow\ L\in\mathcal H. \] Under this interpretation, the proposed statement is \[ \forall\text{ finite groups }G,\qquad \left|\left\{\mathcal H\subseteq\operatorname{form}(G): \begin{array}{l} \mathcal H\text{ is a formation of finite groups},\\ \mathcal H\text{ is subnormal-subgroup-closed} \end{array} \right\}\right|<\infty. \] This explicit formulation is conditional on the interpretation of $S_n$ and the finite-group setting. In particular, it does not treat $n$ as a separately quantified integer or impose a bound on subnormal defect; the supplied record does not settle those notational questions.

Short English statement

The proposed assertion says that the formation generated by each finite group contains only finitely many $S_n$-closed subformations. If $S_n$-closure means closure under subnormal subgroups, the displayed formulation states this explicitly. The meaning of $S_n$ and the ambient setting require confirmation from the source.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $701066$, exact source problem code catalogue entry, titled "Kourovka Notebook Problem 14.81." The supplied registry labels the problem open; this is a source label, not an independently verified mathematical or literature assessment.
• Attributed source: The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), Problem 14.81. Proposed by A. N. Skiba and L. A. Shemetkov.
• The supplied background also identifies the 21st issue edition (2026), arXiv:1401.0300v48. The collection description credits E. I. Khukhro and V. D. Mazurov as editors. No source text defining $S_n$, external URL, or literature-verification date is supplied; the cited work has not been independently consulted for this draft.