I tried many examples , but i can't find any counterexample .
But I guess there are many counter examples , and specific sorts of groups or subgroups have this property (e.g abelian groups or normal subgroups).
Thus I have two question:
- Is there any counter example of group $G$ and its subgroup $H$ s.t there is no surjective homomorphism from $G$ to $H$ ?
- If exists some counterexamples , which sort of groups or subgroups have this property?
I would also appreciate any reference .