# Correspondence between Abelian Groups and Z-Modules/Homomorphisms

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## Theorem

Let $G, H$ be abelian groups.

Let $f : G \to H$ be a mapping.

The following are equivalent:

- $f$ is a group homomorphism.
- $f$ is a $\Z$-module homomorphism between the $\Z$-modules associated with $G$ and $H$.