Kourovka Notebook Problem 15.102

Definitions and mathematical statement

Let $F_n$ denote the free group of rank $n$. The supplied source does not specify the value or range of $n$, or whether its question concerns a fixed rank or existence in some rank; the question below retains this rank parameter.

The commutator subgroup is \[ [F_n,F_n]=\langle x^{-1}y^{-1}xy : x,y\in F_n\rangle. \] For $r\in F_n$, its normal closure in $F_n$ is \[ \langle\!\langle r\rangle\!\rangle_{F_n} =\langle grg^{-1}:g\in F_n\rangle. \] An element $r\in F_n$ is a normal root of $u\in F_n$ if $u\in\langle\!\langle r\rangle\!\rangle_{F_n}$. Two elements $r,s\in F_n$ are conjugate if $s=grg^{-1}$ for some $g\in F_n$.

Question (attributed in the source to W. Magnus). Can there exist an element \[ u\in F_n\setminus[F_n,F_n] \] and an infinite subset $R\subseteq F_n$ such that \[ \begin{gathered} \forall r\in R,\qquad u\in\langle\!\langle r\rangle\!\rangle_{F_n},\\ \forall r,s\in R,\quad r\ne s\ \Longrightarrow\ \left(\forall g\in F_n,\ s\ne grg^{-1}\right)? \end{gathered} \] Thus all normal closures and conjugacy relations are taken inside the same group $F_n$.

Short English statement

Can an element of a free group lying outside its commutator subgroup have infinitely many pairwise nonconjugate normal roots? A normal root is an element whose normal closure contains the given element. The source leaves the rank parameter unspecified.

Sources

• Local UnsolvedMath registry: permanent ID $701190$, exact source problem code catalogue entry, title Kourovka Notebook Problem 15.102; supplied record updated 2026-10-16T14:00:00Z.
• Supplied bibliographic attribution: The Kourovka Notebook (Unsolved Problems in Group Theory), 15th issue (2002), Problem 15.102; 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 source identifications have not been independently verified.
• The record lists V. E. Shpilrain as proposer, with proposed year 2002; its statement separately attributes the question to W. Magnus. Both attributions are retained.
• The registry labels the problem open. This is a supplied status label, not an independently verified assessment of the literature.