Tour:Group of integers modulo n

This article adapts material from the main article: group of integers modulo n

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{{WHAT YOU NEED TO DO:

Understand the definition of the group of integers modulo $n$
Check that the addition tables given for small values of $n$ are correct.}}

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 $(Z / n Z, +)$ or simply $Z / n Z$ . 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.

$n = 2$ :

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

$n = 3$ :

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

$n = 4$ :

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

Definition

Examples

See also