---
tag: 0JM7PA288B
kind: statement
role: theorem
name: Cauchy's theorem
aliases: [cauchys-theorem-group-theory]
msc: [20D20]
uses: [0JPYSD5E0B, 3C53TD, 098D2E7DEH]
sources:
  - url: https://en.wikipedia.org/wiki/Cauchy's_theorem_(group_theory)
    locator: lead section
  - url: https://en.wikipedia.org/wiki/Cauchy's_theorem_(group_theory)
    locator: Statement and proof
same_as:
  wikidata: Q1139041
  wikipedia: Cauchy's theorem (group theory)
---

## Statement

Let $G$ be a finite [[group|G5S1M9]] and let $p$ be a prime number that divides $|G|$.
Then $G$ has an element of [[order|0WE2NHZJ7B]] $p$.

## Remarks

Equivalently, $G$ has a [[subgroup|0X1JXT47GA]] of order $p$, namely the cyclic subgroup generated by such an element.
It is a partial converse of [[Lagrange's theorem|3C53TD]]: a prime divisor of $|G|$ is always the order of a subgroup.
One proof lets $\mathbb{Z}/p\mathbb{Z}$ act by cyclic shifts on the set of tuples $(g_1, \dots, g_p)$ with $g_1 g_2 \cdots g_p = e$, and uses the [[fixed point lemma|098D2E7DEH]].
The [[first Sylow theorem|0V98WY94B8]] generalises the result to prime powers.

## History

Augustin-Louis Cauchy discovered the theorem in 1845.

## Explanations

### L1 human

Lagrange says that the order of an element divides the order of the group.
Cauchy says that, for prime numbers, each divisor really appears as the order of an element.
For example, a group of order $15$ has an element of order $3$ and an element of order $5$.