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burnside-problem
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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.