Definition: A group is called hopfian if every surjective homomorphism is injective. Clearly every finite group is hopfian.
Problem: Prove that is not hopfian.
In the homomorphism you give what is mapped to x, or to odd powers of x? If you hadn't given it a name I would be sceptical such groups existed: in a homomorphic image you add a relation into the presentation, xy=yx or whatever. However, that makes xy and yx equivalent and so you don't map to one of them.
Just a thought...
Interesting. I stand corrected. By rights I should probably stare at this group for the next hour or two, but I actually have to study group rings (this site is wonderful - I can procrastinate by studying maths...). Anyway - to complete the solution:
and as is not trivial we have a non-trivial kernel, and so no injection.