i'm trying to construct an construct an epimorphism from and from but unfortunately i have no idea how to start...
For a in and n in Z, map (a, n) to n. That's trivial.
For what do you mean by " " and " "? My first thought was that " " was the set of permutations on 3 objects and " " was the set of pairs of complex numbers but in that case, is finite (containing 6 members) while is infinite so there cannot be such an epimorphism.
You haven't given enough information. The morphism is an epimorphism for the first one. The morphism is an epimorphism for the second one. I am not sure what more you are trying to do with this. If you want, you can use which would also be an epimorphism for the first one.
An explicit epimorphism , where :
Proving this is a homomorphism is the tricky part, it may be easier to observe that what we are actually doing is mapping:
, which is isomorphic to any cyclic group of order 2, since it is of order 2 (any two groups of prime order are isomorphic and both are cyclic).