# Group of integers modulo n

## Definition

Let $n$ be a positive integer. The group of integers modulo $n$ is an Abelian group defined as follows:

• Its underlying set is the set $\{ 0,1,2,\dots,n-1 \}$
• The rule for addition in the group is as follows. If the integer sum $a + b$ is between $0$ and $n - 1$, then the sum is defined as equal to the integer sum. If the integer sum $a + b$ is at least $n$, then the sum is defined as $a + b - n$.
• The identity element of the group is $0$.
• The inverse map in the group is defined as follows: the additive inverse of $0$ is $0$, and the additive inverse of any other $a$ is, as an integer, $n - a$.

The group of integers modulo $n$ is a concrete description of the cyclic group of order $n$.

This group is typically denoted as $(\mathbb{Z}/n\mathbb{Z},+)$ or simply $\mathbb{Z}/n\mathbb{Z}$. It is also sometimes denoted as $C_n$.

## Examples

Here are the multiplication tables (more aptly called addition tables, because the group is Abelian and the operation is more typically called addition) for the group of integers mod $n$ for small values of $n$:

$n = 1$:

$+$ $0$
$0$ $0$

This is isomorphic to the trivial group.

$n = 2$:

$+$ $0$ $1$
$0$ $0$ $1$
$1$ $1$ $0$

$n = 3$:

$+$ $0$ $1$ $2$
$0$ $0$ $1$ $2$
$1$ $1$ $2$ $0$
$2$ $2$ $0$ $1$

$n = 4$:

$+$ $0$ $1$ $2$ $3$
$0$ $0$ $1$ $2$ $3$
$1$ $1$ $2$ $3$ $0$
$2$ $2$ $3$ $0$ $1$
$3$ $3$ $0$ $1$ $2$