There exist abelian groups whose isomorphism classes of direct powers have any given period
From Groupprops
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)
View a complete list of group property non-implications | View a complete list of group property implications
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 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 .
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 , non-isomorphic groups such that but .
References
- On a conjecture of Pierce concerning direct decomposition of Abelian groups by A.L.S. Corner, Proc. Colloq. Abelian Groups, Page 43 - 48(Year 1964): ^{}^{More info}
- A MathOverflow discussion