Group of units of Z/nZ

Description

Let $n \geq 1$ be an integer. The group of units of $\mathbb{Z}/n\mathbb{Z}$ is the group of units $(\mathbb{Z}/n\mathbb{Z})^\times$ of the ring $\mathbb{Z}/n\mathbb{Z}$ under multiplication. Its elements are the residue classes $[a]$ with $\gcd(a, n) = 1$, and its identity is $[1]$. It is a finite abelian group of order $\varphi(n)$.

Remarks

For $n = 8$, the group is $\{[1], [3], [5], [7]\}$, and the square of each element is $[1]$, so it is isomorphic to the Klein four-group. For $n = 16$, the group is isomorphic to $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/4\mathbb{Z}$, so it is not cyclic. The group is cyclic if and only if $n$ is $1$, $2$, $4$, $p^k$ or $2 p^k$, with $p$ an odd prime and $k \geq 1$. A generator is called a primitive root modulo $n$. Other notations are $(\mathbb{Z}/n\mathbb{Z})^*$, $U(\mathbb{Z}/n\mathbb{Z})$ and $\mathbb{Z}_n^*$.

Explanations

L1 human

Multiplication modulo $n$ does not make all of $\mathbb{Z}/n\mathbb{Z}$ a group, because $[0]$ has no inverse. The classes coprime to $n$ do form a group under multiplication. This group is important in number theory and in cryptography.