I have a function $f: \mathbb Z^4 \to \mathbb Z^2$ defined by: $$f(x,y,u,v):= (x+2y+u,y-3v) $$ I have to show that it's a homomorphism and that $\ker(f)$ is isomorphic to $\mathbb Z^2$.
I showed the first part just by looking at $f((x+x'),(y+y'),(u+u'),(v+v'))$ and got that it was equal to $f(x,y,u,v)+f(x',y',u',v')$ which by definition mean, that it's homomorphic.
But I have trouble showing that $\ker(f)$ is isomorphic to $\mathbb Z^2$, I want to define a homomorphism from the kernel to $\mathbb Z^2$ and show that it's a isomorphism, but don't know how to proceed , any suggestions?
Lastly, What possibilities are there generally for the structure of the kernel for a homomorphism $g: \mathbb Z^4 \to \mathbb Z^4$? Could we use the structure theorem for finitely generated abelian groups maybe?