20.13 (2022)

(a) If an abelian group can be written as a homomorphic image of a nonprincipal countable ultraproduct of not necessarily abelian groups $G_i$, must it be a homomorphic image of a nonprincipal countable ultraproduct of abelian groups? See (G. M. Bergman, Pacific J. Math., 274 (2015), 451–495).

(b) If an abelian group can be written as a homomorphic image of a direct product of an infinite family of not necessarily abelian finite groups, can it be written as a homomorphic image of a direct product of finite abelian groups?

To see that neither question is trivial, choose for each $n > 0$ a finite group $G_n$ which is perfect but has elements which cannot be written as products of fewer than $n$ commutators. Then both the direct product of the $G_n$ and any nonprincipal ultraproduct of those groups will have elements which are not products of commutators; hence its abelianization $A$ will be nontrivial. There is no evident family of abelian groups from which to obtain $A$ as an image of a nonprincipal ultraproduct, nor a family of finite abelian groups from which to obtain $A$ as an image of a direct product.