Kourovka Notebook Problem 14.17

Definitions and mathematical statement

For a group $G$, define $[a,b]=a^{-1}b^{-1}ab$, $G'=[G,G]$, and $G''=[G',G']$. The characteristic subgroup $G''$ determines the second derived quotient $G/G''$. Define \[ \mathrm{IMA}(G)=\ker\!\left(\mathrm{Aut}(G)\longrightarrow\mathrm{Aut}(G/G'')\right) =\{\alpha\in\mathrm{Aut}(G):\alpha(g)G''=gG''\text{ for every }g\in G\}. \] For each integer $n\geq 3$, let $F_n=\langle x_1,\ldots,x_n\rangle$ be the free group of rank $n$.

Problem (V. G. Bardakov, 1999). Find generators and defining relations for $\mathrm{IMA}(F_n)$, for every integer $n\geq 3$. More precisely, describe a generating family $S_n$ of automorphisms in $\mathrm{IMA}(F_n)$ and a family of relations $R_n$ such that the homomorphism determined by these automorphisms gives an isomorphism \[ \langle S_n\mid R_n\rangle\cong\mathrm{IMA}(F_n). \] The supplied statement does not require either family to be finite.

Short English statement

For each free group of finite rank $n\geq 3$, describe generators and all defining relations for the group of automorphisms inducing the identity on $F_n/F_n''$. The supplied registry labels this problem open; that status has not been independently verified.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $701002$, exact source problem code catalogue entry, title Kourovka Notebook Problem 14.17; upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82. The record was updated on 2026-10-16 and carries the supplied status open.
• The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), Problem 14.17, proposed by V. G. Bardakov. The supplied background also identifies the 21st issue edition (2026), arXiv:1401.0300v48, https://arxiv.org/abs/1401.0300v48. The supplied collection description credits E. I. Khukhro and V. D. Mazurov as editors.