Group of units modulo n

Definition

Let $n$ be a positive integer. The group of units modulo $n$ is an abelian group defined as follows:

• Its underlying set is the set $\{a:(a,n)=1\}$
• The group operation is multiplication modulo $n$.
• The identity element of the group is $1$.
• The inverse of an element $a$ in the group is the unique $x$ in the group such that $ax=1$.

This group is typically denoted as $(\mathbb{Z}/n\mathbb{Z},\times {)}^{\times }$ or simply $(\mathbb{Z}/n\mathbb{Z}{)}^{\times }$.