$A=\{z\in\mathbb{C}: z^{p^n}=1\;\mathrm{for\;some\;integer}\;n\geq1\}$. I have to prove:
1) Every proper subgroup of $A$ is cyclic,
2) if $B,C$ are subgroups of $A$ then $B\subseteq C$ or $C\subseteq B$,
3) for every $n\geq0$ there exists an unique subgroup of $A$ with $p^n$ elements.
Could you help me please?