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$F$ is finitely generated free abelian group. Any abelian group homorphism $\phi:F\to C^n$'s image is free where $C$ is complex number?

$\textbf{Q:}$ Is above statement true? I do not see any good reason to generate torsion elements. It is clear that the image is finitely generated abelian group. However, it is not clear whether it has to be free though it seems.

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The image is obviously torsion-free, since $\mathbb{C}^n$ is torsion-free. So since the image is torsion-free and finitely generated, it is free.

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