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I want to study the structure of cokernel of abelian group homomorphism.

Is it true that $\mathbb{Z}^2/<(1,1)>$ is cyclic group isomorphic to $\mathbb{Z}$?

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Let $\varphi\colon\mathbb Z^2\to \mathbb Z$, $\varphi(x,y)=x-y$. Then $\varphi$ is a homomorphism. Since $\ker\varphi = \langle (1,1)\rangle$ and $\varphi$ is surjective, the first isomorphism theorem gives the answer.

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