MHB Is ψ an Isomorphism from H to G?

[himynameJEF]

I'm trying to figure out how to prove this, but I'm unsure how to approach it.

Let G and H be groups, let ϕ: G → H be an isomorphism, and let ψ be the inverse function of ϕ. Prove that ψ is an isomorphism from H to G.

any help? thanks

 

[Olinguito]

Here are some hints to start you off: $\psi$ is injective because $\phi$ is well defined, and it is surjective because $\phi$ is defined on all elements in its domain. There, you just have to translate that into mathematical symbols. (Smile) As for it being a homomorphism, observe that for $x,y\in G$,
$$xy\ =\ \left[\phi(\psi(x))\right]\left[\phi(\psi(y))\right]\ =\ \phi\left[\psi(x)\psi(y)\right].$$
Now apply $\psi$ to both sides.