Suppose that $H$ is a (cyclic) subgroup of order $m$ of a cyclic (abelian) group $G$ of order $n$. What is $G/H$?

Question

Suppose that $H$ is a (cyclic) subgroup of order $m$ of a cyclic (abelian) group $G$ of order $n$. What is $G/H$?

This is taken from an exercise at the end of a section that I must have read 6 or 7 times and still can't relate to this question. I am doing some independent study and am 253 pages into the book and this is the first time that I have fundamentally not understood a topic.

Thank you

Answer 1

$G/H$ is a cyclic group of order $\frac nm$. in fact, a generator of $G$ maps to a generator of $G/H$ under the canonical map.