1. ## Isomorphism

Show that Z*Z/2Z is notisomorphic to Z.(I have to look for order of few elements)

2. Originally Posted by math.dj
Show that Z*Z/2Z is notisomorphic to Z.(I have to look for order of few elements)

I suppose that by Z*Z/2Z you meant the direc product of Z and Z/2Z and not their free product, otherwise it is trivial: the direct product is abelian and the free product isn't.

And anyway it is very easy: the element $(0,a)\in\mathbb{Z}\times \mathbb{Z}\slash 2\mathbb{Z}$ , with $\mathbb{Z}\slash 2\mathbb{Z}=\{1,a\}$ , has finite order and thus must be mapped to the unit in $\mathbb{Z}$ under any homomorphism ...

Tonio