Cyclic subgroups of finite groups Let's say I want to list all the cyclic subgroups of $G$. Let's say $G = \mathbb{Z}^*_{10}$. Then I know all the elements in $G$ are $1, 3, 7, 9$ so all I need know is to find the cyclic subgroups from those elements. As I understand I need to find subgroups so that all elements generate from one element? Then if I'm right the subgroups are $\{1\}, \{3, 9\}, \{7\}, \{9\}$? Is that right?

Perhaps it's easier to note that $\,\Bbb Z_{10}^*\cong C_4=$ the cyclic group of order $\,4\,$, so that there are exactly
three subgroups here: 
$$\{1\}\,,\,\,\{1,9\}\,,\,C_4$$
Check that $\,\{3\}\,,\,\{9\}\,$ cannot be subgroups as they don't contain the unit element...
