[Math] Describe all group homomorphisms from Z×Z into Z


I find a similar post, which is Describe all ring homomorphisms from Z×Z into Z.
I also know the difference between group and ring. But in this case, from ZxZ into Z, I'm so confused. The textbook I'm using has very few examples of group homomorphisms on Z.

How can I solve this problem? Thank you!

Best Answer

For any pair $(a,b)\in \mathbb Z\times\mathbb Z$ you find a group homomorphism by assigning $(x,y)\mapsto ax+by$. Conversely, any group homomorphism $f$ is determined by the values of $(1,0)$ and $(0,1)$. By defining $a=f(1,0)$ and $b=f(0,1)$ it is easy to see that $f$ is precisely of the above form, i.e. equals the map $(x,y)\mapsto ax+by$. Hence these are precisely the group homomorphisms from $\mathbb Z^2$ to $\mathbb Z$. In other words, these group homomorphisms correspond bijectively to the set $\mathbb Z^2$.

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