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 $\displaystyle (0,a)\in\mathbb{Z}\times \mathbb{Z}\slash 2\mathbb{Z}$ , with $\displaystyle \mathbb{Z}\slash 2\mathbb{Z}=\{1,a\}$ , has finite order and thus must be mapped to the unit in $\displaystyle \mathbb{Z}$ under any homomorphism ...
Tonio