Show Surjective Homomorphism but Not Isomorphism

Question

If $G$ and $H$ are two groups, how do I show that the map $\phi:G\times H\to G$ such that $\phi((g,h))=g$ is a surjective homomorphism but not an isomorphism?

Answer 1

An isomorphism is a bijective homomorphism.

Now, $\ker \phi = \{ (1,h) : h \in H \}$ and so $\phi$ is injective iff $H=1$.

Answer 2

If $|H|\geq2$ then $\exists h'\not=h \in H$ such that $\phi((g,h))=\phi((g,h'))=g$ but $(g,h)\not=(g,h')$