Kourovka Notebook Problem 20.124

Definitions and mathematical statement

Let $F$ be a non-abelian free group. The supplied record imposes no restriction on its rank or finite generation. Define its derived subgroup by \[ [F,F]=\langle x^{-1}y^{-1}xy : x,y\in F\rangle. \] A Rota--Baxter operator on $F$ is a mapping $B:F\to F$ satisfying \[ B(g)B(h)=B\left(gB(g)hB(g)^{-1}\right) \qquad\text{for all }g,h\in F. \] The question is whether there exists such an operator with \[ \operatorname{im}(B)=\{B(g):g\in F\}=[F,F]. \] Here $F$ is an arbitrary non-abelian free group, as in the supplied formulation; no particular rank is specified.

Short English statement

Given a non-abelian free group $F$, does there exist a Rota--Baxter operator whose image is exactly the derived subgroup $[F,F]$? The supplied registry labels this problem open.

Sources

• Local UnsolvedMath registry, permanent ID $701782$, source code catalogue entry, title Kourovka Notebook Problem 20.124. The supplied record labels the problem open.
• Attributed in the supplied record to V. G. Bardakov, proposed in 2022.
• Supplied source attribution: The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), problem 20.124; also identified with the 21st issue edition (2026), arXiv:1401.0300v48. The registry describes the collection as edited by E. I. Khukhro and V. D. Mazurov. These bibliographic details are supplied source data and have not been independently verified.