# Epimorphism between abelian groups

I am new to group theory, and I need some help with surjective group homomorphisms for abelian groups.

Let's say I have two finitely generated abelian groups: $$G=\mathbb{Z}^n\oplus \mathbb{Z}_{m_{1}} \oplus \dots \oplus \mathbb{Z}_{m_{s}}$$ and $$H= \mathbb{Z}^t\oplus \mathbb{Z}_{l_{1}} \oplus \dots \oplus \mathbb{Z}_{l_{k}}$$.

Now I want to test if there is a surjective group homomorphism $$\phi: G \to H$$.

Let's say I also have the group $$L=\mathbb{Z}^t$$. I know that there is a surjective group homomorphism $$f: G \to L$$ if $$n\ge t.$$

Is there a similar statement to test if there is a surjective group homomorphism $$\phi$$?

Consider $$\pi \circ \phi : G \to L$$, where $$\pi : H \to L$$ is the natural projection.
If $$\phi$$ is surjective, then so is $$\pi \circ \phi$$, and therefore $$n\ge t$$.