Kourovka Notebook Problem 10.73

Definitions and mathematical statement

A formation of finite groups is a nonempty class $\mathcal F$ of finite groups, closed under isomorphisms, homomorphic images, and finite subdirect products. Here a subdirect product of $G_1,\ldots,G_t$, with $t\geq 1$, is a subgroup of $\prod_{i=1}^t G_i$ whose projection onto each factor is surjective. A subformation of $\mathcal F$ is a subclass $\mathcal H\subseteq\mathcal F$ that is itself a formation; equality is permitted.

The source asks: enumerate all formations of finite groups all of whose subformations are $S_n$-closed. The supplied record does not define $S_n$-closed. Consequently, the following explicit formulation is conditional on interpreting this notation as closure under taking subnormal subgroups; this interpretation requires source verification.

A subgroup $K\leq G$ is subnormal in $G$ if there are an integer $r\geq 0$ and subgroups \[ K=K_0\trianglelefteq K_1\trianglelefteq\cdots\trianglelefteq K_r=G. \] Under the stated interpretation, the problem is to classify all formations $\mathcal F$ of finite groups such that \[ \begin{gathered} \text{for every formation }\mathcal H\subseteq\mathcal F, \text{ every }G\in\mathcal H,\text{ and every }K\leq G,\\ K\text{ subnormal in }G\quad\Longrightarrow\quad K\in\mathcal H. \end{gathered} \] If the source uses a different meaning of $S_n$-closed, this conditional formulation must be revised. In particular, the supplied record does not specify an integer parameter $n$ or a quantifier over such a parameter.

Short English statement

Classify the formations of finite groups for which every subformation has the source's $S_n$-closure property. If this means closure under subnormal subgroups, every subnormal subgroup of every group in each subformation must belong to that same subformation. The meaning of $S_n$-closed remains to be verified.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $700688$, exact source code catalogue entry, titled Kourovka Notebook Problem 10.73. The record attributes the problem to L. A. Shemetkov, proposed in 1986, and labels it open; that label is supplied registry data, not an independently verified mathematical assessment. The record was updated on 2026-10-16T14:00:00Z.
• The Kourovka Notebook (Unsolved Problems in Group Theory), original problem 10.73, 10th issue (1986); the supplied background identifies the 21st issue edition (2026), arXiv:1401.0300v48. The collection description names E. I. Khukhro and V. D. Mazurov as editors and describes coverage of Issues 1--21 (1965--2026). This attribution and edition information are supplied by the record; the cited source has not been independently consulted.