Let $G$ be a abelian group and $H\subseteq G$ is a subgroup. Prove: There exists homomorphism $Q$ such that $\ker Q=H$.

Question

Let $G$ be a abelian group and $H\subseteq G$ is a subgroup. Prove: There exists homomorphism $Q$ such that $\ker Q = H$.

So far I've tried to define homomorphism $Q$ such that $\ker Q = H$ with no success. I'm not even sure how to use the fact that $G$ is abelian.

Any hints will be useful.

Answer 1

Hint : $Q : G \to G/H $

by $Q(g) =g+H$