show that $f:G \ \times H \rightarrow G$ Given by $f((x,y))=x$ is a surjective homomorphism.
I'm not really sure on how to tackle this homework problem.
So I tried with letting H = $f((c,d))=b$
Then $f((x,y),(c,d))=xb=f((x,y)),f((c,d))$ Hence it is a homomorphism?
To show it is onto I have to show that for every cartesian product of G and H there exists an element in G?
In fact, $f((a,b)(c,d))=f((ac,bd))=ac=f((a,b))f((c,d))$, so that it is a homomorphism. To prove it is surjective, only show that every element of $G$ can be represented as the result of this function. Suppose $x\in G$. Therefore, $x=f((x,y)).$ It is also good to note that this is not a isomorphism, once $f^{-1}$ is not well-defined.