The problem
General Burnside problem: is every finitely generated periodic group (all elements of finite order) finite? Restricted version: among exponent-n groups generated by d elements, is there a largest finite one?
The Burnside Problems
openHistory & significance
Burnside asked in 1902. Golod and Shafarevich (1964) answered the general question with spectacular NO: infinite finitely generated groups of bounded exponent exist, built by their hierarchical construction (later powering infinite class-field-tower examples). The restricted problem fell to Efim Zelmanov in 1990 via Lie-algebra methods and the Restricted Burnside algorithm — recognized with the 1994 Fields Medal. One question, two answers, opposite signs, forty years apart.
Still open.
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