Kourovka Notebook Problem 21.11

Definitions and mathematical statement

Write $\omega=\{0,1,2,\ldots\}$. A nonprincipal ultrafilter $\mathcal U$ on $\omega$ is a collection of subsets closed under finite intersections and supersets, containing no finite set, such that exactly one of $A$ and $\omega\setminus A$ belongs to $\mathcal U$ for every $A\subseteq\omega$. For a sequence of groups $(G_i)_{i\in\omega}$, its ultraproduct is \[ \prod_{i\in\omega}G_i/\mathcal U =\left(\prod_{i\in\omega}G_i\right)/ \{(g_i):\{i:g_i=1\}\in\mathcal U\}. \] Multiplication is coordinatewise before passing to this normal-subgroup quotient. A homomorphic image means the target of a surjective abstract group homomorphism, with no continuity requirement.

Determine which groups in the following two classes can occur as such images. Part (a) concerns infinite finitely generated groups of finite exponent: a finite list generates the group, and some integer $n\ge1$ satisfies $g^n=1$ for every element $g$. Part (b) concerns $\operatorname{FSym}(X)$ for an infinite set $X$, the group of all bijections $X\to X$ having finite support $\{x:g(x)\ne x\}$.

For each proposed target, both the sequence of source groups and the nonprincipal ultrafilter may be chosen. Countability refers to the index set of the family; the individual groups $G_i$ need not be countable or finite. The source asks whether some or all members of each target class occur, so existence for a single target and occurrence for every target are distinct parts of the requested determination.

Short English statement

Which infinite finitely generated finite-exponent groups and finitary symmetric groups are quotients of countably indexed nonprincipal ultraproducts?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2520 (KOU-21.11), Kourovka Notebook - New Problems, Issue 21. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 The Kourovka Notebook, 21st edition, new problems, Problem 21.11. Expanded formulation of the supplied catalogue statement. https://arxiv.org/pdf/1401.0300v47.