What's the quotient of the group

Question

Suppose we have an exact sequence of abelian groups:

$0 \rightarrow \mathbb{Z} \rightarrow \mathbb{Z} \oplus \mathbb{Z} \rightarrow H \rightarrow 0$, where the map $\mathbb{Z} \rightarrow \mathbb{Z} \oplus \mathbb{Z}$ is given by $t \rightarrow (2t, -2t).$

What is the solution for group $H$?

Answer 1

Hint: Take $u=(1,1),v=(1,0)\in\mathbb Z\oplus\mathbb Z$. These generate the group, and their images generate $H$. What is the order of each element in $H$?