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Let $\Gamma$ be a totally ordered abelian group such that $\Gamma\cong (\mathbb{Z},+)$ as groups.

Then, how do I prove that there exists a group isomorphism $f:\Gamma\rightarrow (\mathbb{Z},+)$ which sends positive cone to positive cone? (i.e. $x\leq y \Rightarrow f(x)\leq f(y)$)

It's written in wikipedia that a valuation ring whose value group is isomorphic to $(\mathbb{Z},+)$ is a discrete valuation ring, and I am trying to figure out why.

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The isomorphism tells you that $\Gamma$ is generated by a single element $a$. W.l.o.g. $0<a$ (otherwise take $-a$ as generator). Then $na\mapsto n$ gives you the desired isomorphism preserving the ordering.

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