Question about "Quotient Group of Cyclic Group is Cyclic"

Question

I found a proof of the fact that

if $G$ is a cyclic group and $H$ is a subgroup of $G$, then $G/H$ is a cyclic subgroup.

They don't mention that $H$ is a normal subgroup. But to define the quotient group, doesn't $H$ have to be normal?

https://proofwiki.org/wiki/Quotient_Group_of_Cyclic_Group

Answer 1

Cyclic groups are abelian. Let $h\in H$, $g\in G$ for $H\le G$, abelian $G$. Then $$ghg^{-1}=gg^{-1}h=h\in H.$$ Thus $H\unlhd G$.