Multiplicative group of integers modulo n

Definition
This group is defined as the multiplicative group of the ring of integers modulo n, i.e., the group $$(\mathbb{Z}/n\mathbb{Z})^\times$$.

The group is a finite abelian group (though it is not necessarily cyclic) and the order of this group is $$\varphi(n)$$ where $$\varphi$$ denotes the Euler totient function.

Contrast with the additive group of integers modulo n which is just a finite cyclic group of order $$n$$.

Facts

 * Classification of natural numbers for which the multiplicative group is cyclic
 * Characterization of multiplicative group of integers modulo n: This describes the multiplicative group of integers modulo $$n$$ as a direct product of cyclic groups of orders computed explicitly in terms of the prime factorization of $$n$$.
 * Multiplicative group of a prime field is cyclic
 * Multiplicative group of a finite field is cyclic