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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).

notdeleting 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. $\endgroup$ – KConrad Feb 1 '12 at 21:34