---
tag: 0FMAQSB1SC
kind: statement
role: theorem
name: Burnside's p^a q^b theorem
aliases: [burnsides-theorem]
msc: [20D10]
uses: [0P3KZK0FTV, 0C093D1T4M]
sources:
  - url: https://en.wikipedia.org/wiki/Burnside's_theorem
    locator: lead section
  - url: https://en.wikipedia.org/wiki/Burnside's_theorem
    locator: History
same_as:
  wikidata: Q1996305
  wikipedia: Burnside's theorem
---

## Statement

Let $p$ and $q$ be prime numbers, let $a, b \geq 0$ be integers, and let $G$ be a finite [[group|G5S1M9]] of order $p^a q^b$.
Then $G$ is [[solvable|0FXJ7YFMMX]].

## Remarks

A consequence is that the order of each finite [[simple group|0YMYMNQ0QZ]] that is not abelian has at least three distinct prime divisors.
For example, $A_5$ has order $60 = 2^2 \cdot 3 \cdot 5$.
For $b = 0$, the result says that finite $p$-groups are solvable, which follows from [[their nilpotency|081WTKYX6C]].

## History

William Burnside proved the theorem in 1904, with the representation theory of finite groups.
Burnside, Jordan and Frobenius had proved special cases before, in 1897, 1898 and 1902.
John G. Thompson later showed that his work on the N-group theorem gives a proof without representation theory.

## Explanations

### L1 human

If the order of a group has only two prime factors, then the group is solvable.
So a simple group that is not abelian needs at least three primes in its order.
The smallest such group, $A_5$, uses the primes $2$, $3$ and $5$.