Multiplicative groups of braces with finitely generated addition

Definitions and mathematical statement

A brace is a nonempty set $G$ equipped with binary operations $+$ and $\circ$ such that $(G,+)$ is an abelian group, $(G,\circ)$ is a group, and \[ a\circ(b+c)+a=(a\circ b)+(a\circ c)\qquad(a,b,c\in G). \] The additive group is finitely generated if there exist an integer $n\geq 0$ and elements $g_1,\ldots,g_n\in G$ such that every element of $G$ is an integer linear combination of these elements.

For a group $H$, define its derived series by $H^{(0)}=H$ and $H^{(k+1)}=[H^{(k)},H^{(k)}]$, where $[K,K]$ is the subgroup generated by the commutators $x^{-1}y^{-1}xy$ with $x,y\in K$. The group $H$ is solvable if $H^{(m)}=\{1_H\}$ for some integer $m\geq 0$. Let $F_2$ denote the free group on two generators; containing a nonabelian free subgroup is equivalent to containing a subgroup isomorphic to $F_2$.

The two existence questions are:

1. (a) Does there exist a brace $(G;+,\circ)$ such that $(G,+)$ is finitely generated and $(G,\circ)$ is not solvable?
2. (b) Does there exist a brace $(G;+,\circ)$ such that $(G,+)$ is finitely generated and there exists an injective group homomorphism $F_2\longrightarrow(G,\circ)$?

Each part is a separate existence question. The supplied record imposes no additional finiteness or torsion assumptions on $G$.

Short English statement

The problem asks whether a brace with finitely generated abelian additive group can have a multiplicative group that (a) is not solvable, or (b) contains a nonabelian free subgroup. The supplied registry labels the problem open; this status has not been independently verified.

Sources

• Local UnsolvedMath registry, permanent ID $701784$, exact source problem code catalogue entry, title Kourovka Notebook Problem 20.126. The supplied record was created and updated on 2026-10-16T14:00:00Z; its upstream revision is d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• The Kourovka Notebook (Unsolved Problems in Group Theory), Problem 20.126, attributed in the supplied record to V. G. Bardakov, M. V. Neshchadim, and M. K. Yadav, proposed in 2022. The background cites the 20th issue (2022) and the 21st issue edition (2026), identified as arXiv:1401.0300v48.
• The supplied collection description identifies Kourovka Notebook (Issues 1--21), covering 1965--2026 and edited by E. I. Khukhro and V. D. Mazurov. These bibliographic details and the open-status label are supplied source information, not independently verified assessments.