Determine if $\mathbb{Z}\times\mathbb{Z}\to S_4:(1,0)\to(1,2,3),(0,1)\to(2,3,4)$ is a homomorphism

Question

Is it even possible for $\mathbb{Z}\times\mathbb{Z}$ to be homomorphic to $S_4$, since $\mathbb{Z}\times\mathbb{Z}$ is infinite? I know I have to show that $\phi(a\cdot b)=\phi(a)\cdot\phi(b)$, but I don't really understand how the given map works.

Answer 1

Hint: $$(0,1)+(1,0)=(1,0)+(0,1)$$

The actual idea is to assume that there is a homomorphism meeting this description (since, if it exists, then it will be unique). You must then determine if it is well-defined.

Incidentally, there is a homomorphism $\Bbb Z\to S_4.$ For example, $1\mapsto(1,2,3,4)$ works. This is neither one-to-one (obviously) nor onto, but it does yield a unique homomorphism.