Non isomorphic groups who product with Z is isomorphic

Question

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?

(Answer given in the answers to the question. Answers also link to similar questions.) — user1729, May 15 '18 at 14:36

score 3 · Accepted Answer · answered Apr 12 '11 at 20:34

3

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

answered Apr 12 '11 at 20:34

score 0 · Answer 2 · answered Apr 12 '11 at 20:46

0

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

answered Apr 12 '11 at 20:46

A.G

This is not correct. You need $G$ (and $H$) to be abelian for this to work. – user1729 May 15 '18 at 14:37

2 Answers2