Kourovka Notebook Problem 9.61

Definitions and mathematical statement

A variety of groups is the class of all groups satisfying some specified set of group identities. Let $\mathfrak V$ denote the set of all varieties of groups. Write $M(\mathcal V)$ provisionally for the Mal'cev theory of $\mathcal V$ in the sense of the cited work of D. M. Smirnov. The supplied record defines \[ \mathcal V\sim_S\mathcal W \quad\Longleftrightarrow\quad M(\mathcal V)=M(\mathcal W). \] For $\mathcal V\in\mathfrak V$, its full $S$-equivalence class would therefore be \[ [\mathcal V]_S =\{\mathcal W\in\mathfrak V:M(\mathcal W)=M(\mathcal V)\}. \] Source question (D. M. Smirnov, 1984). What is the cardinality of the set of $S$-equivalent varieties of groups?

Unresolved qualifications. The supplied record does not define Mal'cev theory beyond referring to Smirnov's article, so $M$ above is a placeholder for the source-specific invariant, not an independently specified theory. If the intended set is a full equivalence class, the requested quantity is $\lvert[\mathcal V]_S\rvert$; however, the record neither specifies $\mathcal V$ nor states whether the question asks for a common cardinality, the possible cardinalities, or a maximal cardinality. It also does not explicitly require the family of pairwise $S$-equivalent varieties to be a full equivalence class. These choices remain unresolved.

Short English statement

Smirnov asks how many varieties of groups can have the same Mal'cev theory. A precise cardinality question requires the cited definition of that theory and clarification of which equivalence class or family is intended. The supplied registry labels the problem open; this is not an independently verified status assessment.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $700594$, exact source problem code catalogue entry, title Kourovka Notebook Problem 9.61; upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82. The record supplies the status label open.
• The Kourovka Notebook (Unsolved Problems in Group Theory), problem 9.61, proposed by D. M. Smirnov in the 9th issue (1984). 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.
• D. M. Smirnov, Algebra and Logic, volume 22, no. 6 (1983), pp. 492--501, cited in the supplied statement for Mal'cev theory. No article title or further definition is supplied. This citation and the notebook attribution have not been independently checked.