How do you describe the quotient group $\mathbb{Z} \times \mathbb{Z} / \langle (1, 2) \rangle$? It is easy to get it when it is $\mathbb{Z}_n$, but it is difficult when it is $\mathbb{Z}$.
How should I start and solve the problem?
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Define the homomorphism $\phi: \mathbb{Z}^2 \to \mathbb{Z}$ by $\phi((x,y)) = y-2x$. This is a homomorphism because \begin{align*}\phi((x,y)+(u,v)) &=\phi((x+u,y+v)) \\ &= y+v-2(x+u) \\ &= (y-2x) + (v-2u) \\ &= \phi((x,y))+\phi((u,v))\end{align*} Now, the kernel of $\phi$ is the set $\left\{(x,y) \in \mathbb{Z}^2: y-2x = 0\right\} = \langle (1,2) \rangle$, and the image is all of $\mathbb{Z}$, since for any $t \in \mathbb{Z}$, we have $\phi((0,t)) = t$. Thus, the first isomorphism theorem tells us that $\mathbb{Z}^2/\langle(1,2)\rangle \cong \mathbb{Z}$.
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