Quotient_Group_of_Infinite_Cyclic_Group_by_Subgroup

Let $C_n$ be the cyclic group of order $n$. Then $C_n \cong \dfrac {\struct {\Z, +} } {\struct {n \Z, +} } = \dfrac \Z {n \Z}$ where $\Z$ is the additive group of integers $n \Z$ is the additive group of integer multiples $\Z / n \Z$ is the quotient group of $\Z$ by $n \Z$. Thus, every cyclic group is isomorphic to one of $\Z, \dfrac \Z \Z, \dfrac \Z {2 \Z}, \dfrac \Z {3 \Z}, \dfrac \Z {4 \Z}, \ldots$