In an exercise I am doing I have an abelian group $A$ and $a \in A$. I have to find a homomorphism $$ f: \langle a \rangle \to \mathbb{Q}/\mathbb{Z} $$ such that $f(a) \neq 0$ and then extend this map to $$ f': A \to \mathbb{Q}/\mathbb{Z}. $$
I was trying to prove this by considering How to prove $A \cong \langle a \rangle \oplus A / \langle a \rangle $ for abelian group?, but this seems to be not the right approach... I would appreciate explanation on how I can do this. Thank you.