Additive Group of Integer Modulo n
Definition
For an arbitrary natural number $n$, let the set $\mathbb{Z}_{n}$ and the binary operation $+$ be defined as follows.
$$ \mathbb{Z}_{n} = \{ 0, 1, 2, \cdots, n-1 \} \\ a + b = (a + b) \mod n $$
Here $\operatorname{mod}$ is the modulo operation. The binary operation structure $(\mathbb{Z}_{n}, +)$ is called the integer modulo (additive) group. It is denoted simply by $\mathbb{Z}_{n}$.
Explanation
The binary operation structure $(\mathbb{Z}_{n}, +)$ satisfies the definition of a group.
Identity element
The identity element is the integer $0$. For $a \in \mathbb{Z}_{n}$, $$ a + 0 = 0 + a = a $$Inverse
The inverse of $a \in \mathbb{Z}_{n}$ is $n - a$. $$ a + (n - a) = n \mod n = 0 $$Associativity
Associativity holds because $(a \mod n) + (b \mod n) = (a + b) \mod n$ is valid for the modulo operation. $$ (a + b) + c = (a + b) + c $$
Examples
Concrete examples for small $n$ are as follows.
$\mathbb{Z}_{2}$
$$ \begin{array}{c|cc} \mathbb{Z}_{2} & 0 & 1 \\ \hline 0 & 0 & 1 \\ 1 & 1 & 0 \end{array} $$
$\mathbb{Z}_{3}$
$$ \begin{array}{c|ccc} \mathbb{Z}_{3} & 0 & 1 & 2\\ \hline 0 & 0 & 1 & 2\\ 1 & 1 & 2 & 0\\ 2 & 2 & 0 & 1 \end{array} $$
$\mathbb{Z}_{4}$
$$ \begin{array}{c|cccc} \mathbb{Z}_{4} & 0 & 1 & 2 & 3 \\ \hline 0 & 0 & 1 & 2 & 3 \\ 1 & 1 & 2 & 3 & 0 \\ 2 & 2 & 3 & 0 & 1 \\ 3 & 3 & 0 & 1 & 2 \end{array} $$