Kourovka Notebook Problem 13.52

Definitions and mathematical statement

For a finite abelian group $A$, let $\operatorname{Var}(A)$ denote the variety of groups generated by $A$, namely the class of groups satisfying every group identity satisfied by $A$.

For a finitely based variety of algebras $\mathcal V$, the source writes $SC(\mathcal V)$ for the theory consisting of the strong Mal'cev conditions satisfied by $\mathcal V$. A basis of this theory means an independent generating set: it generates the whole theory, and no member follows from the remaining members under the consequence relation used for this theory. Let $\mathscr B(\mathcal V)$ denote the collection of these bases. The source defines the dimension of $\mathcal V$ to be the maximal length of such a basis when these lengths are bounded, and to be infinite when they are unbounded. Thus finite dimension requires a uniform finite bound on the lengths of all bases, not merely the existence of one finite basis.

The question is whether \[ \forall A\;\left(A\text{ is a finite abelian group}\ \Longrightarrow\ \dim(\operatorname{Var}(A))<\infty\right). \] Equivalently, in the source's framework, does every finite abelian group $A$ admit a nonnegative integer $d_A$ such that every basis of $SC(\operatorname{Var}(A))$ has at most $d_A$ members?

The supplied excerpt does not specify the formal syntax or consequence relation for strong Mal'cev conditions, and hence does not fully specify generation and independence in the $SC$-theory. These notions must be read in the sense intended by the cited source; the formulation above retains that unresolved qualification.

Short English statement

Does every finite abelian group generate a variety whose theory of strong Mal'cev conditions has a finite upper bound on the sizes of all its independent generating sets? The bound may depend on the group. The precise consequence relation for this theory is not specified in the supplied excerpt.

Sources

• Local UnsolvedMath registry: supplied complete record with permanent ID $700970$, exact source problem code catalogue entry, title Kourovka Notebook Problem 13.52, and upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• Attributed to D. M. Smirnov, proposed in 1995. The supplied reference is The Kourovka Notebook (Unsolved Problems in Group Theory), 13th issue (1995), Problem 13.52, as included in the 21st issue edition (2026), arXiv:1401.0300v48. The collection description names E. I. Khukhro and V. D. Mazurov as editors.
• The local record labels the problem open. This is supplied registry information, not an independently verified assessment of its current mathematical status. The cited source has not been independently consulted for this draft.