Editing Burnside problem (section)

== Brief history ==

Initial 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 one. This provides a solution for the [[#Restricted Burnside problem|restricted Burnside problem]] for the case of prime exponent. (Later, in 1989, [[Efim Zelmanov]] was able to solve the restricted Burnside problem for an arbitrary exponent.) [[Issai Schur]] had shown in 1911 that any finitely generated periodic group that was a subgroup of the group of invertible ''n'' × ''n'' complex matrices was finite; he used this theorem to prove the [[Jordan–Schur theorem]].<ref name="Curtis">{{cite book |title=Representation Theory of Finite Groups and Associated Algebras |last=Curtis |first=Charles  |author2=Reiner, Irving |year=1962 |publisher=John Wiley & Sons |pages=256–262}}</ref>

Nevertheless, the general answer to the Burnside 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]] supplied a negative solution to the bounded exponent problem for all odd exponents larger than 4381 which was later improved to an odd exponent larger than 665 by Adian.<ref name="Olshanskii">{{cite book |last1=Olʹshanskiĭ |first1=A. I︠U︡ |title=Geometry of defining relations in groups |date=1991 |publisher=Kluwer Academic Publishers |location=Dordrecht ; Boston |isbn=9780792313946 |page=xxii |access-date=26 April 2024 |url=https://books.google.com/books?id=uS_pCAAAQBAJ}}</ref> In 1982, [[A. Yu. Ol'shanskii]] found some striking counterexamples for sufficiently large odd exponents (greater than 10<sup>10</sup>), and supplied a considerably simpler proof based on geometric ideas.

The case of even exponents turned out to be much harder to settle. In 1992, S. V. Ivanov announced the negative solution for sufficiently large even exponents divisible by a large power of 2 (detailed proofs were published in 1994 and occupied some 300 pages). Later joint work of Ol'shanskii and Ivanov established a negative solution to an analogue of the Burnside problem for [[hyperbolic group]]s, provided the exponent is sufficiently large. By contrast, when the exponent is small and different from 2, 3, 4 and 6, very little is known.