Proof that all subgroups of $(\Bbb Z_{15},+)$ are cyclic and list all of its distinct groups

Question

Two questions:

1. Prove that all subgroups of $\Bbb Z_{15}$ are cyclic
2. List all distinct groups of $\Bbb Z_{15}$.

For part 1) I've done this much:

$$\gcd(r,15) = 1$$

The generators are $1,2,4,7,8,11,13,14$

I'm not sure what to do from this point.

Thanks

Answer 1

Any subgroup of any cyclic group is itself cyclic.

By Lagrange's theorem, the subgroups have orders $1,3,5$ or $15$. It is then easy to see that the subgroups are $$\{[0]_{15}\},$$ $$\{[0]_{15}, [5]_{15}, [10]_{15}\},$$ $$\{[0]_{15}, [3]_{15}, [6]_{15}, [9]_{15}, [12]_{15}\},$$ and $\Bbb Z_{15}$, where

$$[a]_{n}=\{b\in\Bbb Z : n\mid a-b\}.$$