subgroups of a factor group

Hi! I have a question concerning subgroups: Define $\displaystyle H_p=\{\frac{n}{p^\alpha}: n, \alpha \in \mathbb{Z}, \alpha \geq 0\}$, where p is a prime. Then $\displaystyle H_p$ is a subgroup of $\displaystyle (\mathbb{Q},+)$. I want to show:

(1) In the quotient group $\displaystyle \mathbb{Q} / \mathbb{Z}$ the subgroup $\displaystyle G_p=H_p / \mathbb{Z}$ is exactly the set of elements with order a power of p (EDIT).

(2) What are the real subgroups of $\displaystyle G_p$?

Does anyone have ideas for how to solve this? I am thankful for any hints.

Banach