Homomorphism homework

[Laila Mohamed]

Hi, I have a homework but I can't solve it. Can someone help me ?

Let ϕ : G → H be a homomorphism of groups.
Show that if ϕ is an isomorphism of groups, then its inverse[math]ϕ^-1 : H → G, ϕ(g) → ϕ^−1 (ϕ(g)) = g[/math], is an isomorphism of groups.

I know that ker(ϕ) = e ⇐⇒ injective, but I don't know how to proceed and I completely don't know how to prove that this is surjective too.

 

[blamocur]

Which definition of isomorphism are you using?
In Wikipedia isomorphism is defined as a bijective homomorphism -- can you prove that the inverse is bijective too?

 

[Steven G]

ϕ^-1(a+b) = a+b = ϕ^-1(a) + ϕ^-1(b).
Continue

 

[blamocur]

Which definition of isomorphism are you using?
In Wikipedia isomorphism is defined as a bijective homomorphism -- can you prove that the inverse is bijective too?

Forgot to add: you also need to prove that the inverse homomorphism exists.