Kourovka Notebook Problem 21.15

Definitions and mathematical statement

For an infinite set $\Omega$, let $S_\Omega$ be the group of all bijections $\Omega\to\Omega$ under composition. Let $B\leq S_\Omega$ be any subgroup, with no closure or measurability assumption. Take two abstract copies of $S_\Omega$ and identify their copies of each $b\in B$. Their amalgamated free product \[ P=S_\Omega*_B S_\Omega \] is the quotient of their ordinary free product by the normal subgroup generated by the relations identifying these two copies of $b$, for all $b\in B$. Its canonical factor maps $j_1,j_2:S_\Omega\to P$ are injective and have intersection exactly the common copy of $B$.

The question asks whether \[ \forall\text{ infinite sets }\Omega\ \forall B\leq S_\Omega\ \exists\text{ injective group homomorphism } \varphi:S_\Omega*_B S_\Omega\longrightarrow S_\Omega. \] This is abstract group embeddability. The map is not required to restrict to the given inclusion of $B$ into $S_\Omega$, and no topological condition is imposed on it in the question.

The supplied solution answers no already when $\Omega$ is countably infinite. In that case $S_\Omega$ carries the pointwise-convergence topology, whose identity neighborhoods fix finite subsets pointwise; it is a Polish group. Choose a subgroup $B$ that is not Borel in this topology. The source attributes to Corson the following counterexample argument. If an embedding $\varphi$ existed, its two factor restrictions $\varphi_i=\varphi\circ j_i$ would be continuous by the cited automatic-continuity result. As continuous injective images of a Polish space, their images would be Borel by the Lusin--Souslin theorem. The amalgam normal-form intersection property gives \[ \varphi(j_1(B))=\operatorname{im}(\varphi_1)\cap\operatorname{im}(\varphi_2), \] so taking the inverse image under $\varphi_1$ would make $B$ Borel, a contradiction. This is the supplied proof summary, not a newly checked result or a construction for every subgroup.

The background says the question weakens the group case of Bergman's 2007 Question 4.4 and that an earlier 2009 result obstructed embeddings required to respect $B$. The current counterexample is stronger in that it allows any abstract embedding and still rules it out for the chosen non-Borel subgroup. A countably infinite instance suffices to refute the universal question over all infinite sets; no conclusion for every uncountable set or every Borel subgroup is inferred.

The registry labels this problem solved. Its literature triage dated 2026-08-17 agrees, attributes the counterexample to Corson's letter of 26 January 2026 and linked solution, and reports no remaining issue for the yes/no question. These mathematical and status statements are attributed to the supplied sources; no external proof was visited or independently verified here.

Short English statement

Must two copies of an infinite symmetric group, amalgamated over an arbitrary subgroup, embed back into that symmetric group? The supplied solution says no: a non-Borel subgroup in the countable case obstructs even an arbitrary abstract embedding via automatic continuity.

Sources

• Kourovka Notebook, New Problems, 21st issue (2026), original Problem 21.15; supplied discussion attributed to G. M. Bergman.
• G. M. Bergman, Indagationes Mathematicae 18 (2007), 349--403, Question 4.4; title not supplied in the record.
• Related $B$-respecting embedding result, Algebra and Number Theory 3 (2009), 847--879, Section 10, as supplied in the Bergman discussion without a title.
• Samuel M. Corson, "Solution of Kourovka Notebook Problem 21.15" (2026), with letter dated 26 January 2026 in the source: https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2026/03/solution-of-21.15.pdf. This citation is retained without opening or generating a PDF.
• Alexander S. Kechris and Christian Rosendal, "Turbulence, amalgamation, and generic automorphisms of homogeneous structures," Proceedings of the London Mathematical Society 94, no. 2 (2007), 302--350: https://arxiv.org/abs/math/0409567.
• Local registry data/unsolvedmath/problems.json, permanent catalogue entry, source code catalogue entry, "Kourovka Notebook Problem 21.15"; statement, complete supplied solution discussion, and literature triage dated 2026-08-17. Dataset attribution: Ulam AI and the UnsolvedMath Contributors.