Burnside's problem

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The Burnside problem, posed by William Burnside in 1902 and one of the oldest and most influential questions in group theory, asks whether a finitely generated group in which every element has finite order must necessarily be a finite group. In plain language, if by looking at individual elements of a group we suspect that the whole group is finite, must it indeed be true? The problem has many variants (see bounded and restricted below) that differ in the additional conditions imposed on the orders of the group elements.


Brief history

Initial work pointed towards the affirmative answer. For example, if a group G is generated by m elements and the order of each element of G is a divisor of 4, then G is finite. Moreover, A. I. Kostrikin (for the case of a prime exponent) and Efim Zelmanov (in general) proved that, among the finite groups with given number of generators and exponent, there exists a largest one. Issai Schur showed that any finitely generated periodic group that was a subgroup of the group of invertible n x n complex matrices was finite; he used this theorem to prove the Jordan–Schur theorem.[1]

Nevertheless, the general answer to Burnside's problem turned out to be negative. In 1964, Golod and Shafarevich constructed an infinite group of Burnside type without assuming that all elements have uniformly bounded order. In 1968, Pyotr Novikov and Sergei Adian's supplied a negative solution to the bounded exponent problem for all odd exponents larger than 4381. In 1982, A. Yu. Ol'shanskii found some striking counterexamples for sufficiently large odd exponents (greater than 1010), and supplied a considerably simpler proof based on geometric ideas.

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