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

Arithmetic functions

Function	Value	Explanation
order of a group	$\varphi (n)$, the Euler totient function of $n$
exponent of a group	$\lambda (n)$, the universal exponent of $n$