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

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 $$d\phi$$.

• Why do you say you have no topological information to use here? You know that $\phi$ is a submersion. Submersions are locally projections, and are thus open maps. – jgon Apr 19 at 14:44
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.
• Thank you @Aristide. Why is $d\phi$ a local isomorphism? it is surjective but is it injective? – PerelMan Apr 19 at 14:58
• $f$ the quotient map $G/N\rightarrow H$ is a local isomorphism. – Tsemo Aristide Apr 19 at 15:00