Prove that a quotient group is cyclic

**crushingyen** · February 8th 2010, 01:34 AM

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 $<x> = G$ . So I would need to show that $<Nx> = G$ .

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

Thank you.

**Swlabr** · February 8th 2010, 02:13 AM

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 $<x> = G$ . So I would need to show that $<Nx> = G$ .

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 $x^n$ 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?

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