---
tag: 0F8X8ZRQ53
kind: object
name: Group of units of Z/nZ
aliases: [multiplicative-group-of-integers-modulo-n]
msc: [11A07, 20K01]
instance_of: [0FJ90Y9926, 082GHNPJ53]
sources:
  - url: https://en.wikipedia.org/wiki/Multiplicative_group_of_integers_modulo_n
    locator: lead section
  - url: https://en.wikipedia.org/wiki/Multiplicative_group_of_integers_modulo_n
    locator: Notation
  - url: https://en.wikipedia.org/wiki/Multiplicative_group_of_integers_modulo_n
    locator: Powers of 2
same_as:
  wikidata: Q1169249
  wikipedia: Multiplicative group of integers modulo n
---

## Description

Let $n \geq 1$ be an integer.
The group of units of $\mathbb{Z}/n\mathbb{Z}$ is the [[group of units|0FJ90Y9926]] $(\mathbb{Z}/n\mathbb{Z})^\times$ of the ring $\mathbb{Z}/n\mathbb{Z}$ under multiplication.
Its elements are the residue classes $[a]$ with $\gcd(a, n) = 1$, and its identity is $[1]$.
It is a finite [[abelian group|082GHNPJ53]] of order $\varphi(n)$.

## Remarks

For $n = 8$, the group is $\{[1], [3], [5], [7]\}$, and the square of each element is $[1]$, so it is isomorphic to the [[Klein four-group|0PMRQJVTME]].
For $n = 16$, the group is isomorphic to $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/4\mathbb{Z}$, so it is not cyclic.
The group is cyclic if and only if $n$ is $1$, $2$, $4$, $p^k$ or $2 p^k$, with $p$ an odd prime and $k \geq 1$. A generator is called a primitive root modulo $n$.
Other notations are $(\mathbb{Z}/n\mathbb{Z})^*$, $U(\mathbb{Z}/n\mathbb{Z})$ and $\mathbb{Z}_n^*$.

## Explanations

### L1 human

Multiplication modulo $n$ does not make all of $\mathbb{Z}/n\mathbb{Z}$ a group, because $[0]$ has no inverse.
The classes coprime to $n$ do form a group under multiplication.
This group is important in number theory and in cryptography.