1
$\begingroup$

Could someone please verify my following proof? I am using (https://math.stackexchange.com/q/1317875)'s idea for the proof, but no one upvoted their answer, so I am not sure whether it is valid.

If group $G$ is cyclic, then quotient group $G/H$ is cyclic.

Let $G=\left \langle g \right \rangle$. Then for all $g'\in G$, there exists an integer $k$ such that $g'=g^{k}$. By definition of $G/H$, $g'H=g^{k}H=(gH)^{k}$ for any $g'H\in G/H$. By definition, $G/H=\left \langle gH \right \rangle$ and so it is cyclic.

$\endgroup$
  • 1
    $\begingroup$ Proof looks good to me $\endgroup$ – leibnewtz Mar 8 '18 at 4:44
  • $\begingroup$ The proof is correct $\endgroup$ – Mikhail Goltvanitsa Mar 8 '18 at 6:21
1
$\begingroup$

Your proof is fine and generalizes to this:

If $\phi: G \to \Gamma$ is a group homomorphism and $G$ is a cyclic, then the image of $\phi$ is cyclic.

We can then take $\Gamma = G/H$ and $\phi$ the canonical projection $g \mapsto gH$, which is surjective.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.