Find all cyclic subgroups of $\mathbb{Z}_{24}$.

Question

Find all cyclic subgroups of $\mathbb{Z}_{24}$.

I know the subgroups of $\mathbb{Z}_{24}$ to be $\mathbb{Z}_{24}, 2\mathbb{Z}_{24}, 3\mathbb{Z}_{24}, 4\mathbb{Z}_{24}, 6\mathbb{Z}_{24}, 8\mathbb{Z}_{24}, 12\mathbb{Z}_{24}, 24\mathbb{Z}_{24}=\langle0\rangle$.

Which of these are cyclic and why?

Answer 1

As $\mathbb{Z}_{24}$ is cyclic (it is generated by the element $1$), and a subgroup of a cyclic subgroup is cyclic, all the subgroups are cyclic.