Group of integers modulo n

From Groupprops
Jump to: navigation, search

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