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?

Observe your "groups":
A set cannot be a subgroup unless it also contains the identity element of the original group!
Recall:
$H\le (G,*) \iff $:
$H$ is closed under $*$,
The identity of $G$ is IN $H$.
$H$ is closed under inversion. (For all $h \in H, h^{-1} \in H$).
