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.