20.10 (2022)

(a) If $\mathscr{U}$ and $\mathscr{U}'$ are nonprincipal ultrafilters on $\mathbb{N}$, can every group which can be written as a homomorphic image of an ultraproduct of groups with respect to $\mathscr{U}$ also be written as a homomorphic image of an ultraproduct of groups with respect to $\mathscr{U}'$ ?

(b) If the answer to (a) is negative, is it at least true that for any two nonprincipal ultrafilters $\mathscr{U}$ and $\mathscr{U}'$ on $\mathbb{N}$, there exists a nonprincipal ultrafilter $\mathscr{U}''$ on $\mathbb{N}$ such that every group which can be written as a homomorphic image of an ultraproduct of groups with respect to $\mathscr{U}$ or with respect to $\mathscr{U}'$ can be written as a homomorphic image of an ultraproduct with respect to $\mathscr{U}''$?

A positive answer to (b) would imply that the class of groups which can be written as homomorphic images of nonprincipal countable ultraproducts of groups is closed under finite direct products. See (G. M. Bergman, Pacific J. Math., 274 (2015), 451–495).