Kourovka Notebook Problem 7.45

Definitions and mathematical statement

Let $\mathcal F$ denote the class of all finite groups. A group $G$ is finitely presented if there exist nonnegative integers $m,r$ and words $w_1,\ldots,w_r$ in the free group $F_m$ such that \[ G\cong F_m/\langle\!\langle w_1,\ldots,w_r\rangle\!\rangle, \] where the denominator is the normal subgroup generated by the displayed words.

Write $T_Q(\mathcal K)$ for the $Q$-theory of a class $\mathcal K$ in the specific sense of Kourovka Notebook Problem 2.40. This is an unresolved source-defined term: the supplied record does not include Problem 2.40 or define $Q$-theory, so its logical fragment and semantics cannot be specified from the supplied evidence.

The question attributed to D. M. Smirnov is whether \[ \exists\,G\quad\left(G\text{ is a finitely presented group}\ \land\ T_Q(\mathcal F)=T_Q(\{G\})\right). \] This preserves the requested equality as a statement schema; a fully specified mathematical formulation requires the definition referenced in Problem 2.40. No alternative meaning of $Q$-theory is assumed.

Short English statement

Does a single finitely presented group have exactly the same $Q$-theory as the class of all finite groups, using the meaning of $Q$-theory specified in Kourovka Notebook Problem 2.40? That definition is absent from the supplied record.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $700441$, exact source problem code catalogue entry, title Kourovka Notebook Problem 7.45; upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82. The record was created and updated on 2026-10-16 and labels the problem open; this is supplied registry metadata, not an independently verified mathematical or literature assessment.
• The Kourovka Notebook (Unsolved Problems in Group Theory), Problem 7.45, proposed by D. M. Smirnov in the 7th issue (1980). The supplied record identifies the 21st issue edition (2026), arXiv:1401.0300v48. The collection description names E. I. Khukhro and V. D. Mazurov as editors.
• The statement expressly refers to Kourovka Notebook Problem 2.40 for the meaning of $Q$-theory. Its text is not supplied, and neither that definition nor the cited edition has been independently consulted for this draft.