By an argument which is not entirely trivial we know that if $G,H$ are finite groups and $G \times G \cong H \times H$, then $G \cong H$. I was wondering if this result holds if $G, H$ are infinite, and if not, then what a counterexample is.
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According to the wikipedia entry Rank of an abelian group there are rank 2 abelian groups $A=A_{m,n}$ and $B=B_{m,n}$ ($m,n\in\mathbb{N}$) such that $A^n$ is isomorphic to $B^n$ if and only if $m$ divides $n$. If this is so, then of course taking $m=n=2$ gives a counterexample. This result appears to be due to A.L.S. Corner who has various striking results that show that non-finitely generated abelian groups behave very strangely.
Brian M. Scott
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Shane O Rourke
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I've found the paper where this result appears: Corner, A. L. S. On endomorphism rings of primary abelian groups. Quart. J. Math. Oxford Ser. (2) 20 1969 277–296. – Shane O Rourke Apr 05 '12 at 21:25