# Multiplicative group modulo n

From Groupprops

## Definition

Let be a positive integer. The **multiplicative group modulo ** 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 that are relatively prime to . (The two definitions are equivalent because if and are relatively prime, there exist integers such that , so ).

## Facts

- The order of the multiplicative group modulo equals the number of elements in that are relatively prime to . This number is termed the Euler-phi function or Euler totient function of , and is denoted .
- For a prime , . In other words, every nonzero element less than is invertible modulo .
- The multiplicative group modulo is a cyclic group if and only if for an odd prime and a natural number.