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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 $\phi$?

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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 $n\ge t$.

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  • $\begingroup$ This is what the OP already knows. $\endgroup$ – egreg Feb 13 at 15:33

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