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 bounded and restricted below) that differ in the additional conditions imposed on the orders of the group elements, some of which are still open questions.Brief historyInitial work pointed towards the affirmative answer. For example, if a group G is finitely generated and the order of each element of G is a divisor of 4, then G is finite. Moreover, A. I. Kostrikin was able to prove in 1958 that among the finite groups with a given number of generators and a given prime exponent, there exists a largest
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