Kourovka Notebook Problem 19.5

Definitions and mathematical statement

Let $I$ be an infinite index set, and let $(A_i)_{i\in I}$ be a family of finitely generated abelian groups. Let $H$ be a group equipped with injective homomorphisms $\iota_i:H\hookrightarrow A_i$ for every $i\in I$. Write \[ G=\mathop{\ast}_{H,\,i\in I} A_i \] for their free product with amalgamation, identifying the copies $\iota_i(H)$ of $H$. Identify $H$ with its canonical image in $G$. Since $H$ embeds in an abelian group, it is abelian, and its torsion subgroup is \[ \operatorname{Tor}(H)=\{h\in H:\text{there exists an integer }n\geq 1\text{ with }h^n=1\}. \] The source asks whether, for every such amalgamated free product, \[ \psi(G)=\operatorname{Tor}(H). \] However, the supplied record does not define $\psi(G)$. Thus this equality records the intended question only schematically; its precise mathematical meaning requires the source's definition of $\psi$. No countability restriction on $I$, or properness condition on the embeddings of $H$, is stated in the supplied record.

Short English statement

M. K. Azarian asks whether, for a free product of infinitely many finitely generated abelian groups amalgamating a common subgroup $H$, the object denoted by $\psi(G)$ equals the torsion subgroup of $H$. The supplied record leaves $\psi$ undefined, so the question requires clarification before it is a fully specified mathematical statement.

Sources

• Local UnsolvedMath registry, permanent ID $701553$, exact source problem code catalogue entry, title "Kourovka Notebook Problem 19.5." The supplied registry labels the problem as open; this label is not independently verified here.
• Supplied attribution: M. K. Azarian, proposed in 2018; The Kourovka Notebook (Unsolved Problems in Group Theory), original problem number 19.5. The supplied background names the 19th issue (2018) and the 21st issue edition (2026), with identifier arXiv:1401.0300v48. The collection description names E. I. Khukhro and V. D. Mazurov as editors and describes coverage of Issues 1--21 (1965--2026). These bibliographic details are supplied attribution, not independently checked source contents.
• Supplied registry update date: 2026-10-16T14:00:00Z; upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82. No source URL, literature-source links, or literature-verification date is supplied.