# Correspondence between Abelian Groups and Z-Modules/Homomorphisms

## Theorem

Let $G, H$ be abelian groups.

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

The following are equivalent:

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