Kourovka Notebook Problem 19.46

Definitions and mathematical statement

Let $G$ denote the group written $\mathrm{CT}(\mathbb{Z})$ in the supplied source. The source refers to Kourovka Notebook Problem 17.57 for its definition, but that definition is absent from the supplied record. Thus the concrete meaning of $G$ remains unresolved here.

A group $H$ is periodic if every element has finite order, that is, if $\forall h\in H\;\exists n\in\mathbb{Z}_{\geq 1}\; h^n=e_H$. It is finitely generated if $H=\langle h_1,\ldots,h_r\rangle$ for some positive integer $r$ and elements $h_1,\ldots,h_r\in H$.

The question attributed to S. Kohl (2018) is: does there exist a subgroup $H\leq G$ such that $H$ is infinite, finitely generated, and periodic? Equivalently, is there an infinite subgroup $H\leq G$ satisfying \[ \left(\exists r\in\mathbb{Z}_{\geq 1}\;\exists h_1,\ldots,h_r\in H:\ H=\langle h_1,\ldots,h_r\rangle\right) \quad\text{and}\quad \left(\forall h\in H\;\exists n\in\mathbb{Z}_{\geq 1}:\ h^n=e_H\right)? \] The exponent $n$ may depend on $h$; a common finite exponent is not required. This formulation requires the missing definition from Problem 17.57 to identify the ambient group.

Short English statement

Does the source-designated group $\mathrm{CT}(\mathbb{Z})$ contain an infinite subgroup generated by finitely many elements in which every element has finite order? The supplied record omits the referenced definition of the ambient group.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $701594$, exact source code catalogue entry, titled Kourovka Notebook Problem 19.46. Its status is supplied as open; this is not an independently verified assessment.
• The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), Problem 19.46; also identified in the supplied record with the 21st issue edition (2026), arXiv:1401.0300v48. Proposed by S. Kohl in 2018. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors of Issues 1--21 (1965--2026).
• Kourovka Notebook Problem 17.57 is explicitly referenced by the statement for the definition of $\mathrm{CT}(\mathbb{Z})$. Its contents are not supplied and have not been independently consulted.