Let phi psi be maps from Z12 to Z8 Z6 resp defined by phix

Let phi, psi be maps from Z12 to Z8 (Z6, resp.) defined by phi(x) = 3x (mod 8), psi(x) = 3x (mod 6), resp. Determine if psi, psi are well-defined maps, homomorphisms, epimorphisms, monomorphisms, and ismorphisms Hint: One of them is not well-defined

Solution

The first map (phi) is not well-defined. 12 and 24 are equivalent mod 12, but phi(12)=36=4 and phi(24)=72=0. These are not the same mod 8 so phi is not well-defined.

The second map (psi) is well-defined. It is a homomorphism as psi(a+b)=3*(a+b)=3a+3b=psi(a)+psi(b). psi is not onto (cannot get anything except 3 and 0) so it is not an epimorphism. It is also not 1-1 (psi(3)=psi(9), etc.) so it is not a monomorphism. Thus it is certainly not an isomorphism.