AKG, a perfect explanation! Incredible! What Hurkyl and Mathwonk said now make so much sense it isn't funny. Also if I had realised all this, I may have come up with the same idea as Hurkyl and Mathwonk did. I cant believe this came down to prime numbers! Amazing.
This also means that if there are homomorphisms from [itex]\mathbb{Z}_4[/itex] to {0,2} then they are not welldefined. Therefore the only homomorphisms are the ones from {0,1,2,3} > {0,1,2,3} ({0,1,2,3} is a trivial retract) and from {0,1,2,3} > {0} ({0} is a trivial retract).
One last question:
Is it possible to have a countable group with countably many retracts?
I was thinking I would start with finite groups, because they are finitely generated. And every finitely generated group is countable. Then if I want to take a quotient I know that it will be finitely generated as well but Im not sure if I want to do this, but it seems handy to have.
Does this sound like the right place to start?
