Kourovka Notebook Problem 19.96

Definitions and mathematical statement

Let $r\geq 2$ be a finite integer, let $F_r$ be the free group on $r$ generators, and write $F_r'=[F_r,F_r]$ and $F_r''=[F_r',F_r']$. Let $M\cong F_r/F_r''$, so that $M$ is the free metabelian group of rank $r$. A metabelian group is a group whose commutator subgroup is abelian.

An ordered tuple $(b_1,\ldots,b_r)\in M^r$ is a basis of $M$ if the homomorphism $F_r/F_r''\to M$ sending the standard generators to $b_1,\ldots,b_r$ is an isomorphism. Define \[ \mathcal B_r(M)=\{(b_1,\ldots,b_r)\in M^r:(b_1,\ldots,b_r)\text{ is a basis of }M\}. \] An element is primitive if it belongs to a basis. Thus define \[ P(M)=\{g\in M:\exists b_2,\ldots,b_r\in M\text{ with }(g,b_2,\ldots,b_r)\in\mathcal B_r(M)\}. \] The existential quantifier in this displayed definition is an ordinary mathematical quantifier; it does not assume that $\mathcal B_r(M)$ is first-order definable.

Questions (E. I. Timoshenko, Kourovka Notebook Problem 19.96).

1. (a) Is $P(M)$ a first-order definable subset of $M$?
2. (b) Is $\mathcal B_r(M)$ a first-order definable subset of $M^r$?

Here the standard group language is $\mathcal L_{\mathrm{grp}}=\{\cdot,{}^{-1},1\}$, with variables ranging over elements of $M$. Under this language convention, definability in (a) means the existence of a formula $\varphi(x;\mathbf z)$ and an admissible finite parameter tuple $\mathbf a$ from $M$ such that \[ \forall g\in M,\qquad g\in P(M)\ \Longleftrightarrow\ M\models\varphi(g;\mathbf a). \] In (b), it means the existence of a formula $\psi(x_1,\ldots,x_r;\mathbf w)$ and an admissible finite parameter tuple $\mathbf c$ from $M$ such that \[ \forall(b_1,\ldots,b_r)\in M^r,\qquad (b_1,\ldots,b_r)\in\mathcal B_r(M) \ \Longleftrightarrow\ M\models\psi(b_1,\ldots,b_r;\mathbf c). \] Unresolved qualifications. The supplied statement does not explicitly specify the first-order language or whether parameters are allowed. The formulas above explain the standard group-language interpretation without resolving the parameter convention: for parameter-free definability the parameter tuples must be empty, whereas definability with parameters permits finite tuples from $M$. Part (b) is represented by ordered basis tuples; the corresponding family is invariant under permutation of its coordinates. The source does not explicitly specify this encoding or impose a uniformity requirement as the rank varies.

Short English statement

For a free metabelian group of finite rank $r\geq 2$, E. I. Timoshenko asks whether its primitive elements and its bases can each be described by a first-order formula. Bases are represented here by ordered $r$-tuples. The source leaves the language, parameter convention, and any uniformity across ranks unspecified.

Sources

• Local UnsolvedMath registry, permanent ID $701644$, exact source problem code catalogue entry, title Kourovka Notebook Problem 19.96. The supplied record attributes the problem to E. I. Timoshenko, with proposed year 2018.
• Supplied source attribution: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), Problem 19.96; also identified in the record with 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 verified here.
• The supplied registry labels the problem open and gives an update timestamp of 2026-10-16T14:00:00Z. This is source metadata, not an independently verified assessment of the mathematical status.