Why does the image of $ord_p$ form an additive subgroup of $(1/n)\mathbb Z$?

Question

Let $K$ be a field extension of the p-adic rationals $\mathbb Q_p$.
The image of $K^\times$ under the valuation map $ord_p(x)=-\frac 1 n\log_p|\mathbb N_{K/\mathbb Q_p}(x)|_p$ is contained in $(1/n)\mathbb Z = \{x ∈ Q| nx ∈ \mathbb Z\}$, where $n=[K:\mathbb Q_p]$.

The image is an additive subgroup of $(1/n)\mathbb Z$ so it’s of the form $(1/e)\mathbb Z$ for an integer $e$ dividing $n$.

Why is that right? What do we know about the existence of, say, an inverse to any of these subgroup's elements?

Answer 1

To get this off the "unanswered" list, this comment by user metamorphy (May 14, 2019) answers it:

The image consists of all numbers $ord_p(x), x \in K^\times$ by definition - as $xy \in K^\times, x^{−1} \in K^\times$, etc. for $x,y \in K^\times$, it will automatically contain $ord_p(xy), ord_p(x^{−1})$ and the like.