Kourovka Notebook Problem 17.59

Definitions and mathematical statement

Let $\operatorname{Sym}({\mathbb{Z}})$ denote the group of bijections from ${\mathbb{Z}}$ to itself, and put ${\mathbb{Z}}_{\geq 0}=\{0,1,2,\ldots\}$. A permutation $f\in\operatorname{Sym}({\mathbb{Z}})$ is residue-class-wise affine if there exist an integer $m\geq 1$ and integers $a_r\neq 0$, $b_r$, for each $r\in\{0,\ldots,m-1\}$, such that \[ f(r+mk)=a_rk+b_r\qquad\text{for every }k\in{\mathbb{Z}}. \] This expresses the usual meaning of affine on each residue class: equivalently, on that class the formula is $f(n)=(a_r/m)n+b_r-a_rr/m$, with rational coefficients. Bijectivity is required globally.

Define \[ G_+=\left\{f\in\operatorname{Sym}({\mathbb{Z}}): f\text{ is residue-class-wise affine and }f({\mathbb{Z}}_{\geq0})={\mathbb{Z}}_{\geq0}\right\}. \] The setwise condition does not require the nonnegative integers to be fixed individually.

The supplied question, proposed by S. Kohl in 2010 as Kourovka Notebook Problem 17.59, asks whether \[ \mathrm{CT}({\mathbb{Z}})=G_+. \] The supplied record does not define $\mathrm{CT}({\mathbb{Z}})$. Thus this equality preserves the stated target, but a complete mathematical formulation requires the intended definition of its left-hand side. The registry labels the problem open; that label is supplied metadata, not an independently verified assessment.

Short English statement

S. Kohl asks whether $\mathrm{CT}({\mathbb{Z}})$ consists exactly of the permutations of the integers that are affine on each residue class for some common positive modulus and preserve the nonnegative integers setwise. The supplied record leaves $\mathrm{CT}({\mathbb{Z}})$ undefined, so the intended group must be specified before the question is fully precise.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $701361$, exact source problem code catalogue entry, titled Kourovka Notebook Problem 17.59. The supplied status is open.
• Attributed source: The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), Problem 17.59; proposed by S. Kohl in 2010. The background also identifies the 21st issue edition (2026), arXiv:1401.0300v48.
• The supplied collection description attributes the notebook, Issues 1--21 (1965--2026), to editors E. I. Khukhro and V. D. Mazurov. These bibliographic details and the status are supplied attribution and have not been independently verified here.