After studying the group of roots of unity, we now turn to a second fundamental finite example: residue classes modulo $n$ under addition. These groups are among the most important examples in algebra and number theory. They show how arithmetic can be performed inside a finite set while still satisfying all group axioms. The case $\mathbb{Z}_3$ is small enough to compute completely, but the same proof works for every positive integer $n$. In this lesson, students will prove that $(\mathbb{Z}_n,+_n)$ is a finite group.

Let $n\in\mathbb{N}$ with $n\geq2$. For $a\in\mathbb{Z}$, the $\textbf{residue class}$ of $a$ modulo $n$ is denoted by $\overline{a}$ and is defined by $$\begin{equation*} \overline{a}=\{a+kn:k\in\mathbb{Z}\}. \end{equation*}$$ The set of all residue classes modulo $n$ is $$\begin{equation*} \mathbb{Z}_n=\{\overline{0},\overline{1},\ldots,\overline{n-1}\}. \end{equation*}$$