# induced morphism between lie groups is surjective

Let $$φ : G → H$$ be a continous morphism of linear Lie groups with H connected.

Prove that $$φ$$ is surjective iff $$dφ$$ surjective

Using the expression : $$exp\circ dφ = φ \circ exp$$ we can see that

$$φ$$ is surjective $$\Rightarrow φ \circ exp = exp\circ dφ$$ is surjective but how to prove $$dφ$$ is surjective?

I have no idea on how to tackle the other implication.

## Edit

Thanks to comments I see now that

$$dφ$$ is surjective $$\Rightarrow dφ(Lie(g))=Lie(h) \Rightarrow exp\circ dφ(Lie(g))=\exp\circ Lie(h) \Rightarrow φ\circ exp(Lie(g))=H$$ (because H is connected) $$\Rightarrow φ$$ is surjective.

• do you know that a connected Lie group is generated by any neighborhood of the identity? – Tim kinsella Apr 9 at 12:07
• Thanks @Timkinsella I edited the question with your suggestion – PerelMan Apr 9 at 12:31