Kourovka Notebook Problem 12.81

Definitions and mathematical statement

Let $p$ be a prime. A variety of groups is a class of groups defined by group identities. A subvariety of a variety $\mathcal V$ is a variety $\mathcal W$ satisfying $\mathcal W\subseteq\mathcal V$. Write \[ \mathfrak A_p=\{G:G\text{ is abelian and }g^p=1\text{ for every }g\in G\}. \] Thus the exponent divides $p$; the trivial group is included, as required for a variety. No finiteness assumption is imposed.

For varieties $\mathcal U$ and $\mathcal V$, their product $\mathcal U\mathcal V$ consists of the groups $G$ having a normal subgroup $N\trianglelefteq G$ with $N\in\mathcal U$ and $G/N\in\mathcal V$. Under the usual interpretation of the superscript as a product of varieties, put \[ \mathfrak A_p^3=(\mathfrak A_p\mathfrak A_p)\mathfrak A_p, \qquad \operatorname{Sub}(\mathfrak A_p^3) =\{\mathcal W:\mathcal W\text{ is a variety of groups and }\mathcal W\subseteq\mathfrak A_p^3\}. \] Problem. For each prime $p$, determine the cardinality \[ \left|\operatorname{Sub}(\mathfrak A_p^3)\right|. \] The supplied statement does not explicitly define the superscript $3$. The formulation above uses the threefold product convention, which requires confirmation against the original source. The source's phrase "prime exponent $p$" is interpreted by the identity $x^p=1$, rather than by excluding the trivial group.

Short English statement

For a prime $p$, determine how many distinct subvarieties the threefold product of the variety of abelian groups of exponent dividing $p$ has. The product interpretation of $\mathfrak A_p^3$ requires source confirmation. The supplied registry labels the problem open; this draft does not independently verify that status.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $700899$, exact source problem code catalogue entry, title Kourovka Notebook Problem 12.81. The record was created and updated at 2026-10-16T14:00:00Z and carries the status open; these are supplied registry data.
• The Kourovka Notebook (Unsolved Problems in Group Theory), problem 12.81, 12th issue (1992); the supplied background also cites 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 attribution and have not been independently checked.
• Proposed in 1992. The proposer field is malformed; its exact supplied text is V. I. Sushchanski\textbackslash{}u{\textbackslash{}i. No repair of the name is asserted.