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 $\{a:(a,n)=1,0<a<n\}$
• 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 $(\mathbb {Z} /n\mathbb {Z} )^{\times }$.

The order of the group is the Euler totient function evaluated at $n$, $\varphi (n)$.

It is the group of units of a monoid of the monoid of integers modulo n under multiplication.

Results

$(\mathbb {Z} /n\mathbb {Z} )^{\times }$ is cyclic if and only if $n=2,4,p^{k}$ or $2p^{k}$ for $p$ prime, $k$ an integer. See the following pages for part of the proof:

• Group of units modulo prime power is cyclic
  • Group of units modulo prime is cyclic, a special case of the above
• Group of units modulo two times prime power is cyclic

See also

• Group of integers modulo n