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.

$n=2$:

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

$n=3$:

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

$n=4$:

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

See also

• Group of units modulo n