Value group of a valuation

Definition

Let $v:K\to\Gamma\cup\{\infty\}$ be a valuation on a field. Its value group is the ordered abelian subgroup

The group operation records multiplication:

and the order is inherited from $\Gamma$.

Replacing the original codomain by $\Gamma_v\cup\{\infty\}$ makes the valuation surjective without changing its valuation ring or residue field. Thus the value group is intrinsic to the valued field, whereas a larger chosen codomain need not be.