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Stably isomorphic groups

If $G$ and $H$ are two groups (finitely presented, if you wish) with the property that $G\times\mathbb{Z}$ is isomorphic to $H\times\mathbb{Z}$, does that imply that $G$ is isomorphic to $H$?

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This question gets asked so often! –  Steve D Apr 19 '12 at 22:45
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marked as duplicate by Benjamin Steinberg, S. Carnahan Apr 20 '12 at 3:13

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up vote 1 down vote accepted

See Stably isomorphic groups (Hirshon's example is finitely presented).

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