Quotient of additive group of p-adic integers by group of integers

Definition
This group is defined as the quotient group $$\mathbb{Z}_{(p)}/\mathbb{Z}$$, where $$\mathbb{Z}_{(p)}$$ is the additive group of p-adic integers and $$\mathbb{Z}$$ is the group of integers sitting inside it in the obvious way.