The quotient of totally ordered additive abelian groups which is torsion (value groups in valuation theory)

Question

Let $(G,+,\leq)$ be a totally ordered additive abelian group and $(H,+\leq)$ be a subgroup of $(G,+,\leq)$. Let the quotient $\frac{G}{H}$ be torsion and be of finite order $n$ ($|\frac{G}{H}|=n$), where $n$ is a power of a prime number $p\geq 2$ (It is noted that we come across such groups in valuation theory and p-adic number theory as those are the value groups associated to valued fields or local fields). Now, I have the following question. How can we write $G$ according to $H$? Is it true to say $G=\frac{1}{n}H$? In special case, if $H=\mathbb{Z}$, is it true to write $G=\frac{1}{n}\mathbb{Z}$? I saw in Proposition 2.6 of "On saturated distinguished chains over a local field II, J. Number Theory" ( doi.org/10.1016/j.jnt.2004.12.005 ) that $G=\frac{1}{n}\mathbb{Z}-\frac{p}{n}\mathbb{Z}$! But, I don't know its reason. I appreciate any help.