Isomorphic nth powers not implies isomorphic
Suppose . Then, we can find groups such that (i..e, the direct powers are isomorphic groups) but . In fact, we can choose both and to be countable torsion-free (i.e., aperiodic) abelian groups.
- There exist abelian groups whose isomorphism classes of direct powers have any given period: This states that for any , we can find a (countable torsion-free) abelian group such that .
Pick from fact (1), let be the group as arising from fact (1), and let . Then, Since , we obtain that . However, since , . This gives the required example.