TITLE: Non isomorphic groups who product with Z is isomorphic
QUESTION [10 upvotes]: 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?

REPLY [3 votes]: Yes, there is an example in a paper by Hirshon, "On Cancellation in Groups".  You can find a slightly different example written up here