# Prove that a quotient group is cyclic

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

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

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 $\displaystyle G$ is a cyclic group, prove that $\displaystyle G/N$ is cyclic, where $\displaystyle N$ is any subgroup of $\displaystyle G$.

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

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

What does this tell you?