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 $$n$$:

$$n = 1$$:

This is isomorphic to the trivial group.

$$n = 2$$:

$$n = 3$$:

$$n = 4$$: