Unsolved problems in mathematics | Group theory

Burnside problem

The Burnside problem asks whether a finitely generated group in which every element has finite order must necessarily be a finite group. It was posed by William Burnside in 1902, making it one of the oldest questions in group theory and was influential in the development of combinatorial group theory. It is known to have a negative answer in general, as Evgeny Golod and Igor Shafarevich provided a counter-example in 1964. The problem has many refinements and variants (see and below) that differ in the additional conditions imposed on the orders of the group elements, some of which are still open questions. (Wikipedia).

Burnside problem
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Word problem for groups | Order (group theory) | Jordan–Schur theorem | Combinatorial group theory | Index of a subgroup | Finitely generated group | Prüfer group | Group isomorphism | Hyperbolic group | Finite group | Conjugacy problem | Direct product of groups | Dihedral group | P-group | Igor Shafarevich | Tarski monster group | Normal subgroup | Group theory | Cyclic group | Lie algebra | Prime number | Group homomorphism | Issai Schur | Golod–Shafarevich theorem | Presentation of a group