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 $(\mathbb{Z}/n\mathbb{Z},+)$ 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.

$n = 2$:

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

$n = 3$:

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

$n = 4$:

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

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