Kourovka Notebook Problem 18.26

Definitions and mathematical statement

Use the standard finite-group convention that a formation is a nonempty class of finite groups, closed under isomorphisms, homomorphic images, and finite subdirect products. Here a subdirect product of groups $H_1,\ldots,H_n$ is a subgroup of $H_1\times\cdots\times H_n$ whose projection onto each factor is surjective. Write $\mathrm{form}(G)$ for the smallest formation containing $G$. A subformation of $\mathrm{form}(G)$ is a formation contained in that class. These conventions interpret terminology not explicitly defined in the supplied record and require confirmation against the source.

For a prime $p$, an elementary abelian $p$-group is a group isomorphic to $({\mathbb{Z}}/p{\mathbb{Z}})^d$ for some integer $d\geq 0$. A minimal normal subgroup of a group $H$ is a nonidentity normal subgroup containing no proper nonidentity subgroup normal in $H$. Let $A_5$ denote the alternating group on five letters.

The question attributed to V. P. Burichenko is the following. For every prime $p$ and every finite group $G$ admitting normal subgroups $G_1,G_2$ with \[ 1<G_1<G_2<G, \] assume that $G_1$ and $G_2/G_1$ are elementary abelian groups for the same prime $p$, that $G_1$ is a minimal normal subgroup of $G$, that $G_2/G_1$ is a minimal normal subgroup of $G/G_1$, and that \[ G/G_2\cong A_5\times A_5. \] Must the set \[ \left\{\mathcal H:\mathcal H\text{ is a formation and }\mathcal H\subseteq\mathrm{form}(G)\right\} \] be finite?

Short English statement

Does every finite group with two successive elementary abelian layers for the same prime, each minimal normal in the indicated quotient, and quotient $A_5\times A_5$ generate a formation having only finitely many subformations? The supplied registry labels this problem open; that status has not been independently verified.

Sources

• Local UnsolvedMath registry, permanent ID $701455$, exact source code catalogue entry, title "Kourovka Notebook Problem 18.26." The supplied record attributes the problem to V. P. Burichenko, with proposed year 2014, and labels it open.
• The Kourovka Notebook (Unsolved Problems in Group Theory), Problem 18.26, 18th issue (2014); the supplied background also identifies the 21st issue edition (2026), arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors. These bibliographic details are supplied attribution and have not been independently checked.