# Burnside problem

- kind: problem
- status: refuted (evidence tier V3)
- year: 1902
- authors: William Burnside

**Statement.** Is a finitely generated group in which every element has finite order necessarily finite?

No: infinite such groups exist, even with a uniform bound on the orders. The restricted version, which asks only about finite quotients, is true, and Efim Zelmanov's proof of it won a Fields Medal in 1994.

*Evidence tier V3 is the import default for a result the curation records as settled in the published record. No formal proof or certificate is attached to this page yet.*

Posed 1902, William Burnside. Closed 1964 Golod–Shafarevich (general); 1968 Novikov–Adian (bounded exponent); restricted version solved by Zelmanov, 1991.

## Sources

- [Wikipedia: Burnside problem](https://en.wikipedia.org/wiki/Burnside_problem)

[Algebra and groups](/wiki/Space:algebra.md) · [Mathematics](/wiki/Continent:mathematics.md)

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