Cyclic group of order n

Example

Let $n$ be a positive integer. The integers modulo $n$, $\mathbb{Z}/n\mathbb{Z} = \{0, 1, \ldots, n - 1\}$, with addition modulo $n$, form a cyclic group of order $n$. The residue $1$ generates it, because each residue $k$ is the sum of $k$ copies of $1$. Each cyclic group of order $n$ is isomorphic to $\mathbb{Z}/n\mathbb{Z}$, and this group is often written $C_n$.

Remarks

A residue $k$ generates $\mathbb{Z}/n\mathbb{Z}$ exactly when $\gcd(k, n) = 1$. The group of $n$th roots of unity is a cyclic group of order $n$ in multiplicative notation. The rotations of a regular $n$-gon also form a cyclic group of order $n$.

Explanations

L1 human

Think of a clock with $n$ hours, where you add hours and wrap around after $n$. This clock arithmetic gives the cyclic group of order $n$. Every cyclic group of order $n$ looks the same as this clock, up to the names of the elements.