Kourovka Notebook Problem 17.96

Definitions and mathematical statement

A variety of groups is a class consisting of all groups satisfying a specified set of group identities. A group identity is an equation between group words that holds for every assignment of its variables to elements of the group. A subvariety of a variety $\mathcal V$ is a variety of groups contained in $\mathcal V$. Write \[ \operatorname{Sub}(\mathcal V)=\{\mathcal W: \mathcal W\text{ is a variety of groups and }\mathcal W\subseteq\mathcal V\}. \] Does there exist a variety of groups $\mathcal V$ such that \[ \lvert\operatorname{Sub}(\mathcal V)\rvert=\aleph_0 \] and there is a sequence $(\mathcal V_n)_{n\geq 0}$ of subvarieties of $\mathcal V$ satisfying \[ \mathcal V_{n+1}\subsetneq\mathcal V_n\qquad\text{for every integer }n\geq 0? \] Here proper containment is containment of classes of groups. The required chain already forces infinitely many distinct subvarieties, so interpreting "only countably many" as either at most countably many or countably infinitely many gives the same existence question.

Short English statement

Does some variety of groups have exactly countably infinitely many subvarieties while admitting an infinite strictly descending sequence of subvarieties?

Sources

• Supplied local UnsolvedMath registry record, permanent ID $701398$, source code catalogue entry, titled "Kourovka Notebook Problem 17.96."
• The supplied record attributes Problem 17.96 to P. M. Neumann, proposed in 2010, in The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), with reference also to the 21st issue edition (2026), arXiv:1401.0300v48. The collection description names E. I. Khukhro and V. D. Mazurov as editors.
• The registry labels the problem "open." This is a supplied status label, not an independently verified assessment of the literature. The cited source has not been independently consulted for this draft.