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Are there groups $G$ and $H$ such that $G$ and $H$ are not isomorphic but $G \times \mathbb Z$ and $H \times \mathbb Z$ are?

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up vote 3 down vote accepted

Yes, there is an example in a paper by Hirshon, "On Cancellation in Groups". You can find a slightly different example written up here

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Note that if one of them is abelian and finitely generated then by structure theorem $G$ and $H$ are isomorphic.

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