21.13 (2026)

It is known that the group $\mathbb{Z}^\omega$ has a subgroup whose dual is free abelian of rank $2^{\aleph_0}$ (see 17.24 in Archive). Does $\mathbb{Z}^\omega$ have a subgroup whose dual is free abelian of still larger rank (the largest possible being $2^{2^{\aleph_0}}$)? This is Question 11 in (G. M. Bergman, Portugaliae Math., 69 (2012) 69–84).