So I was wondering what are sufficient conditions $\phi$ has to fulfill to be a group homomorphism/isomorphism.
I know that when we consider cyclic groups, $\phi(x)$ is an isomorphism $\iff$ you the generator to another generator (Obviously the groups need to have the same order). Now for homomorphism, we just have to map the generator to an element whose order divides the order of the generator and then we have a homomorphism.
However, things get more complicated when we consider non-cyclic groups, or at least I think. I know what are necessary conditions for a homomorphism/isomorphism are, but not what sufficient conditions are. So I was wondering if there are any theorems regarding this.