Let $\mathbb{Q}/ \mathbb{Z}$ and for $p$ prime let $$S(p)= \{ q+\mathbb{Z} \in \mathbb{Q}/ \mathbb{Z} : \exists k \in \mathbb{N}, k(q+\mathbb{Z}) \} =\mathbb{Z}$$ the $p$- Sylow subgroup in $\mathbb{Q}/ \mathbb{Z}$ show that:
1.- $\forall k \in \mathbb{N}$ there exist unique cyclic subgroups $C_{p^k}$ of order $p^k$ in $S(p)$ such that $$C_p \subset C_{p^2} \subset \ldots \subset S(p)$$ 2.- $S(p)$ is divisible
for $(2)$ i prove first that $\mathbb{Q}/ \mathbb{Z}$ is divisible then in particular for any $q+\mathbb{Z} \in S(p)$ and $\forall k \in \mathbb{N}$ exists $x+\mathbb{Z} \in \mathbb{Q}/\mathbb{Z}$ such that $$k(x+\mathbb{Z})=q+\mathbb{Z}$$ and how $q+\mathbb{Z} \in S(p)$ then exists $m \in \mathbb{N}$ such that $p^m(q+\mathbb{Z})=\mathbb{Z}$ then these implies that $$p^mk(x+\mathbb{Z})=p^m(q+\mathbb{Z})=\mathbb{Z}$$ but I'm not quite sure how to conclude that $x+\mathbb{Z} \in S(p)$.
for (1) i try induction and using the fact that $\mathbb{Q}\mathbb{Z}=\oplus S(p)$ but i think to these is not correct.
Any hint or help i will be very grateful