Burnside's Problem - Brief History

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.

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.

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 Burnside's problem for hyperbolic groups, 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.