Multiplicative group modulo n

Definition

Let $n$ be a positive integer. The multiplicative group modulo $n$ is the subgroup of the multiplicative monoid modulo n comprising the elements that have inverses.

Equivalently, it is the group, under multiplication, of elements in $\{ 0,1,2,\dots,n-1\}$ that are relatively prime to $n$. (The two definitions are equivalent because if $a$ and $n$ are relatively prime, there exist integers $x,y$ such that $ax + ny = 1$, so $ax \equiv 1 \mod n$).

Facts

1. The order of the multiplicative group modulo $n$ equals the number of elements in $\{ 0,1,2,\dots, n-1\}$ that are relatively prime to $n$. This number is termed the Euler-phi function or Euler totient function of $n$, and is denoted $\varphi(n)$.
2. For a prime $p$, $\varphi(p) = p - 1$. In other words, every nonzero element less than $p$ is invertible modulo $p$.
3. The multiplicative group modulo $n$ is a cyclic group if and only if $n = 2,4,p^k,2p^k$ for $p$ an odd prime and $k$ a natural number.