Prove that a quotient group is cyclic

• Feb 8th 2010, 01:34 AM
crushingyen
Prove that a quotient group is cyclic
If $G$ is a cyclic group, prove that $G/N$ is cyclic, where $N$ is any subgroup of $G$.

I first started out stating that since $G$ is a cyclic group, there exists a generator $x \epsilon G$ such that $ = G$. So I would need to show that $ = G$.

I don't know where exactly to continue from this point to finish the proof. Any ideas on how?

Thank you.
• Feb 8th 2010, 02:13 AM
Swlabr
Quote:

Originally Posted by crushingyen
If $G$ is a cyclic group, prove that $G/N$ is cyclic, where $N$ is any subgroup of $G$.

I first started out stating that since $G$ is a cyclic group, there exists a generator $x \epsilon G$ such that $ = G$. So I would need to show that $ = G$.

I don't know where exactly to continue from this point to finish the proof. Any ideas on how?

Thank you.

Well, every elements of your group is of the form $x^n$ for some $n \in \mathbb{Z}$, so every element of your quotient group looks like $x^nN$ for some $n \in \mathbb{Z}$.

What does this tell you?