Kourovka Notebook Problem 16.92

Definitions and mathematical statement

Let $X$ be an arbitrary infinite set, let $F=F(X)$ be the free group with free basis $X$, and put $G=\operatorname{Aut}(F)$, the group of all automorphisms of $F$. The derived subgroup of $G$ is \[ G'=[G,G]=\langle \alpha\beta\alpha^{-1}\beta^{-1}:\alpha,\beta\in G\rangle. \] A group is called perfect if it equals its derived subgroup.

Question. Is $\operatorname{Aut}(F(X))$ perfect for every infinite set $X$? Equivalently, does \[ \operatorname{Aut}(F(X))=[\operatorname{Aut}(F(X)),\operatorname{Aut}(F(X))] \] hold for every infinitely generated free group $F(X)$?

The supplied source states that the equality holds when $X$ is countably infinite, attributing this result to R. Bryant and V. A. Roman'kov, J. Algebra 209 (1998), 713--723. Thus the remaining question concerns arbitrary uncountable ranks. This countable-rank result and the registry's open-status label are supplied source assertions, not independently verified assessments.

Short English statement

Must the automorphism group of every infinitely generated free group be perfect, meaning that every automorphism is a finite product of commutators of automorphisms? The supplied source reports an affirmative answer for countably infinite rank; the question remains directed at uncountable ranks.

Sources

• Local UnsolvedMath registry, permanent ID $701283$, exact source code catalogue entry, title Kourovka Notebook Problem 16.92. The supplied record labels the problem open and records V. Tolstykh as proposer, with proposed year 2006.
• The Kourovka Notebook (Unsolved Problems in Group Theory), Problem 16.92, proposed by V. Tolstykh; 16th issue (2006), also identified in the supplied record with the 21st issue edition (2026), arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors.
• R. Bryant and V. A. Roman'kov, J. Algebra 209 (1998), 713--723. This reference is supplied for the countably generated case; no article title is supplied, and the reference and its claimed scope have not been independently checked.