---
tag: 0XCYGCCX8V
kind: statement
role: theorem
name: Number of elements of each order in a cyclic group
msc: [20K01]
uses: [0B85Q41RP6, 0VVPQNVM8S, 0423TXR9AN]
sources:
  - url: https://en.wikipedia.org/wiki/Cyclic_group
    locator: Additional properties
---

## Statement

Let $G$ be a [[cyclic group|0KVGH0J5B3]] of finite order $n$, and let $d$ be a positive divisor of $n$.
Then $G$ has exactly $\varphi(d)$ elements of [[order|0WE2NHZJ7B]] $d$, where $\varphi$ is Euler's totient function.

## Remarks

Here $\varphi(d)$ is the number of integers $k$ with $1 \leq k \leq d$ and $\gcd(k, d) = 1$.
The elements of order $d$ are the generators of the unique subgroup of order $d$.
Counting all elements by their orders gives the identity $\sum_{d \mid n} \varphi(d) = n$.
In particular, $G$ has exactly $\varphi(n)$ generators.

## Explanations

### L1 human

In $\mathbb{Z}/6\mathbb{Z}$, the element $0$ has order $1$ and the element $3$ has order $2$. The elements $2$ and $4$ have order $3$, and $1$ and $5$ have order $6$.
These counts are $1, 1, 2, 2$, which are $\varphi(1), \varphi(2), \varphi(3), \varphi(6)$.
The totient function counts the elements of each order.