Is the following abelian group injective?

Question

We know that an abelian group is injective iff it is divisible. So $\mathbb Q$ is injective. Now, let $M$ be an abelian group. Is $\operatorname{Hom}(M,\mathbb Q)$ injective?

Answer 1

Let $n \in \mathbb{N}$ and $f \in \text{Hom}(M,\mathbb{Q})$. Define $g(m) = \frac{1}{n} f(m)$ for $m \in M$. Then $g \in \text{Hom}(M,\mathbb{Q})$ and $n g = f$. As $n$ and $f$ were arbitrary, $\text{Hom}(M,\mathbb{Q})$ is divisible, hence injective.