---
tag: FQZJKY2G5Q
kind: statement
role: theorem
name: Restricted Burnside problem
msc: ["20F50", "20D15", "20F40"]
uses: [082DWWXMRJ, 0ZDW0H0VX4, 0CB0DX5BVV, 0DTXBQPA2S]
about: [0ZDW0H0VX4, XWEMJ7T2XV]
sources:
  - url: https://en.wikipedia.org/wiki/Burnside_problem
    locator: Restricted Burnside problem
---

## Statement

Let $m \ge 1$ and $n \ge 1$ be integers.
Then there are only finitely many finite groups, up to isomorphism, that are generated by $m$ elements and have [[exponent|0ZDW0H0VX4]] dividing $n$.
Equivalently, the free Burnside group $B(m, n)$ has only finitely many [[normal subgroups|0CB0DX5BVV]] of finite [[index|0DTXBQPA2S]].
Then the quotient $R(m, n)$ of $B(m, n)$ by their intersection is finite.
Every finite $m$-generated group of exponent dividing $n$ is a quotient of it.

## Remarks

Kostrikin proved the case where $n$ is prime.
Hall and Higman reduced the general case to prime powers $n = p^k$.
Their reduction assumes the Schreier conjecture on outer automorphism groups of finite simple groups, which follows from the classification of finite simple groups.
Zelmanov solved the prime power case around 1990, for which he received a Fields Medal.
The proofs pass to a Lie ring built from a central series of the group, which satisfies Engel identities.
They show that such Lie rings are nilpotent under the right finiteness conditions.

Example: for $n = 2$, every group of exponent $2$ is abelian, and $R(m, 2) = (\Z/2)^m$ has order $2^m$.

The restriction to finite groups is necessary: the group $B(m, n)$ itself is infinite for $m \ge 2$ and many large $n$, by the negative answer to the bounded [[Burnside problem|XWEMJ7T2XV]].

## Explanations

### L1 human

The Burnside problem asks whether a group with finitely many generators, in which $x^n = 1$ for every $x$, must be finite.
The answer is often no.
The restricted version only looks at the finite groups of this kind.
It says that they cannot be arbitrarily large: the number of generators and the exponent bound their order.

### L2 human

The theorem gives a largest finite group $R(m, n)$ with $m$ generators and exponent dividing $n$, and all others are its quotients.
It is a statement about finite groups, but its proof goes through Lie rings.
Kostrikin and Zelmanov proved that Lie algebras satisfying an Engel condition are locally nilpotent under suitable hypotheses.
It contrasts with the bounded [[Burnside problem|XWEMJ7T2XV]], where the free Burnside group can be infinite while all its finite quotients have bounded order.