Integers modulo n form a cyclic group

Example

Let $n \ge 1$. The integers modulo $n$, $\mathbb{Z}/n\mathbb{Z} = \{0, 1, \dots, n-1\}$, form a group under addition modulo $n$. Each element $k$ is the sum $1 + 1 + \dots + 1$ of $k$ copies of $1$. Thus $1$ is a generator, and the group is cyclic of order $n$. Addition modulo $n$ is commutative, so the group is abelian. This agrees with the result that cyclic groups are abelian.