Kourovka Notebook Problem 16.93

Definitions and mathematical statement

Let $n\in{\mathbb{Z}}$ with $n\geq 2$, and let $F_n$ be a free group of rank $n$. Write $A_n=\operatorname{Aut}(F_n)$. For $g\in F_n$, define the inner automorphism $c_g$ by $c_g(x)=gxg^{-1}$ for every $x\in F_n$, and put \[ I_n=\operatorname{Inn}(F_n)=\{c_g:g\in F_n\}\leq A_n. \] Work in the first-order language of groups $\{\cdot,{}^{-1},1\}$. A subset $D\subseteq A_n$ is definable with parameters if there exist a nonnegative integer $m$, a tuple $\bar a\in A_n^m$, and a first-order formula $\varphi(x,\bar y)$ such that \[ \forall\alpha\in A_n,\qquad \alpha\in D\ \Longleftrightarrow\ A_n\models\varphi(\alpha,\bar a). \] Parameter-free definability requires $m=0$.

The question attributed to V. Tolstykh is: for every finite rank $n\geq 2$, is $I_n$ a first-order definable subgroup of $A_n$? The source does not specify whether parameters are permitted, so the parameter-free and parameter-allowed readings remain unresolved qualifications of this draft. No requirement that one formula work uniformly for all ranks is stated.

An element $p\in F_n$ is primitive if it belongs to some free basis of $F_n$. The source additionally states that the set \[ P_n=\{c_{p^k}:p\in F_n\text{ is primitive},\ k\in{\mathbb{Z}}\} \] is definable in $A_n$. Here powers are interpreted as integer powers, including the zeroth power. This auxiliary definability assertion is supplied background; its parameter convention and a supporting proof or separate citation are not supplied.

Short English statement

For each finite rank $n\geq 2$, V. Tolstykh asks whether the inner automorphisms can be defined by a first-order formula inside the group $\operatorname{Aut}(F_n)$. The source leaves the use of parameters unspecified and reports that inner automorphisms induced by powers of primitive elements form a definable set.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $701284$, exact source problem code catalogue entry, title Kourovka Notebook Problem 16.93; upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• The Kourovka Notebook (Unsolved Problems in Group Theory), problem 16.93, proposed by V. Tolstykh in the 16th issue (2006). The supplied background also identifies the 21st issue edition (2026), arXiv identifier 1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors.
• The registry labels this problem open. This is a supplied status label, not an independently verified assessment. The cited edition and the auxiliary definability assertion have not been independently checked in preparing this statement-only draft.