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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The result was proved by A.L.S. Corner in this paper.
Suppose 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 (?)) such that, for any positive integers , we have:
In words, the and direct powers of are Isomorphic groups (?) if and only if and are congruent modulo .
- Isomorphic nth powers not implies isomorphic: We can use this to construct, for any , non-isomorphic groups such that but .