There exist abelian groups whose isomorphism classes of direct powers have any given period

History
The result was proved by A.L.S. Corner in this paper.

Statement
Suppose $$r$$ is a positive integer. Then, there exists a countable fact about::torsion-free abelian group (i.e., an fact about::abelian group that is also a fact about::torsion-free group) $$G$$ such that, for any positive integers $$m,n$$, we have:

$$G^m \cong G^n \iff m \equiv n \pmod r$$

In words, the $$m^{th}$$ and $$n^{th}$$ direct powers of $$G$$ are fact about::isomorphic groups if and only if $$m$$ and $$n$$ are congruent modulo $$r$$.

In particular, we can have an abelian group that is a fact about::group isomorphic to its cube but not a fact about::group isomorphic to its square.

Applications

 * Isomorphic nth powers not implies isomorphic: We can use this to construct, for any $$n > 1$$, non-isomorphic groups $$G,H$$ such that $$G^n \cong H^n$$ but $$G \not \cong H$$.