$\operatorname{coker}(\phi)$ is discrete for a morphism of Lie groups

Question

suppose $\phi: G \to H$ a morphism of Lie groups such that $d\phi$ is surjective. Prove that $\operatorname{coker}\phi$ is discrete.

My attempts:

• Prove that $\phi(G)$ is open which will lead to $H/\phi(G)$ is discrete: we have no topological information to use here.

• prove that $Lie(H/\phi(G))=0$.

I don't know how to proceed and I don't see how to use the surjectivity of $d\phi$.

Thank you for your help.

Answer 1

Let $N$ be the kernel of $\phi$, $G/N$ is a Lie group and there exists a Lie hmomorphism $f:G/N\rightarrow H$, the differential of $f$ at any point of $G/N$ is an isomorphism, the local inverse mapping theorem implies that $f$ is open.