Cyclic subgroup of a cyclic group

Question

Is there any non trivial example of subgroup of a cyclic group?

Every subgroup of a cyclic group is cyclic. I couldnt find any non trivial examples of it since every subgroup would have the generator and if the generator is there then the entire group is there.

Answer 1

Not every subgroup has to contain the generator. Consider the cyclic group of order $4$, which we denote as $\{1,a,a^2,a^3\}$. Then $\{1,a^2\}$ is a proper subgroup.

More generally, if $\langle a \rangle$ is a cyclic group of order $x$ and $d$ divides $x$, we have a cyclic subgroup $\langle a^d \rangle$.

Answer 2

If your group is $G=(\mathbb{Z},+)$, your generator is $1$. If your subgroup is $H=(2\mathbb{Z},+)$, the generator is $2$ which is not the generator for $G$.