Is f a Surjective and Injective Isomorphism from HxN to HN in G?

[mathmajor2013]

Let G be a group, H a normal subgroup, N a normal subgroup, and H intersect N = {e}. Let H x N be the direct product of H and N. Prove that f: HxN->G given by f((h,n))=hn is an isomorphism from HxN to the subgroup HN of G.
Hint: For all h in H and n in N, hn=nh.

 

[micromass]

This looks a lot like a homework/coursework question. This really should belong there instead of the main forums...

To the point: what have you tried already?

 

[mathmajor2013]

I am confused how to start this problem. To first show it is a homomorphism, is f((h,n)(h',n'))=f((hh',nn'))?

 

[micromass]

I am confused how to start this problem. To first show it is a homomorphism, is f((h,n)(h',n'))=f((hh',nn'))?

Yes, that step is already correct. Now apply the definition of f...

 

[mathmajor2013]

Right the homomorphism part is easy now. Am I able to use the pigeonhole principle for the isomorphic part? That is, are HxN and HN the same size? It seems like they are since H intersect N is only the identity.

 

[micromass]

It might be tricky to see that they are thesame size. Isn't easier to show that f is surjective and injective?