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when is A isomorphic to A^3?

Does there exist a group $G$ such that $G \cong G \times G \times G$ and $G \not \cong G \times G$? If such groups exist, can $G$ be countable?

Tangentially, it is known that there is no such linear order (replacing direct product with concatenation) and that there are such Boolean algebras (replacing direct product with direct sum).

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Asher M. Kach
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    Duplicate question: http://mathoverflow.net/questions/10128/when-is-a-isomorphic-to-a3. The answer is yes. – Joel David Hamkins Feb 01 '12 at 20:40
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    Although it has been asked before, let me add, Asher, that it is a great question! It's just that it seems best to keep the answers and discussion all together in one place. – Joel David Hamkins Feb 01 '12 at 21:00
  • Thanks! And agreed. Is there any reason not to delete my question? – Asher M. Kach Feb 01 '12 at 21:03
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    A mild argument for not deleting this question: when I tried to find a question about this theme earlier this week using mathoverflow, I never found the original question because there the notation is A, not G. – KConrad Feb 01 '12 at 21:34

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