1. ## subgroup

Q: list all subgroup of a cyclic group of order 30.

give the hint to do this.

2. The subgroups of any finite cyclic group can be determined the same way:

There is exactly one cyclic subgroup of every order which divides the order of the group.

Of course, this makes answering the question easy: the subgroups of $\mathbb{Z}_{30}$ are $\mathbb{Z}_{30},\mathbb{Z}_{15},\mathbb{Z}_{10},\m athbb{Z}_6,\mathbb{Z}_5,\mathbb{Z}_3,\mathbb{Z}_2, \mathbb{Z}_1=\{ e\}$.

But see if you can prove the statement above on WHY the subgroups of a cyclic group look this way.

3. Originally Posted by topspin1617
Of course, this makes answering the question easy: the subgroups of $\mathbb{Z}_{30}$ are $\mathbb{Z}_{30},\mathbb{Z}_{15},\mathbb{Z}_{10},\m athbb{Z}_6,\mathbb{Z}_5,\mathbb{Z}_3,\mathbb{Z}_2, \mathbb{Z}_1=\{ e\}$.
Embedded isomorphically...