21.12 (2026)

Suppose that $\mathscr{U}$ is a nonprincipal ultrafilter on $\omega$, and $B$ is a group such that every element $b \in B$ belongs to a subgroup of $B$ that is a homomorphic image of $\mathbb{Z}^\omega / \mathscr{U}$. Must $B$ then be a homomorphic image of an ultraproduct group $\prod_{i \in \omega} G_i / \mathscr{U}$ for some groups $G_i$?

This is Question 19 in (G. M. Bergman, Pacific J. Math., 274 (2015) 451–495). An affirmative answer would imply that every torsion group was such a homomorphic image for every $\mathscr{U}$, and so would give positive answers to both parts of 21.11.