Z modulo n/Representatives/Associativity and group/Cyclic/Example

Let ${}n\in \mathbb {N} _{+}$, and consider on

${}\mathbb {Z} /(n)=\{0,1,\ldots ,n-1\}\,$

the binary operation

${}a+b:=(a+b)\mod n={\begin{cases}a+b,{\text{ if }}a+b<n\,,\\a+b-n,{\text{ if }}a+b\geq n\,.\end{cases}}\,$

With this operation, we have a group due to exercise. Because we can write every element as a certain sum of ${}1$ with itself, this is a cyclic group.