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

This article gives the statement and possibly, proof, of a non-implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., group isomorphic to its cube) need not satisfy the second group property (i.e., group isomorphic to its square)
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Get more facts about group isomorphic to its cube|Get more facts about group isomorphic to its square

## 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 Torsion-free abelian group (?) (i.e., an Abelian group (?) that is also a 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 Isomorphic groups (?) if and only if $m$ and $n$ are congruent modulo $r$.

In particular, we can have an abelian group that is a Group isomorphic to its cube (?) but not a Group isomorphic to its square (?).

## Related facts

### 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 $G \not \cong H$.