Burnside problem

The problem has many refinements and variants that differ in the additional conditions imposed on the orders of the group elements (see bounded and restricted below).

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.

In 1964, Golod and Shafarevich constructed an infinite group of Burnside type without assuming that all elements have uniformly bounded order.

Ol'shanskii found some striking counterexamples for sufficiently large odd exponents (greater than 1010), and supplied a considerably simpler proof based on geometric ideas.

Later joint work of Ol'shanskii and Ivanov established a negative solution to an analogue of the Burnside problem for hyperbolic groups, provided the exponent is sufficiently large.

A group G is called periodic (or torsion) if every element has finite order; in other words, for each g in G, there exists some positive integer n such that gn = 1.

Part of the difficulty with the general Burnside problem is that the requirements of being finitely generated and periodic give very little information about the possible structure of a group.

Therefore, we pose more requirements on G. Consider a periodic group G with the additional property that there exists a least integer n such that for all g in G, gn = 1.

Novikov–Adian, Ivanov and Lysënok established considerably more precise results on the structure of the free Burnside groups.

Moreover, the word and conjugacy problems were shown to be effectively solvable in B(m, n) both for the cases of odd and even exponents n. A famous class of counterexamples to the Burnside problem is formed by finitely generated non-cyclic infinite groups in which every nontrivial proper subgroup is a finite cyclic group, the so-called Tarski Monsters.

In terms of category theory, B0(m, n) is the coproduct of n cyclic groups of order m in the category of finite groups of exponent n. In the case of the prime exponent p, this problem was extensively studied by A. I. Kostrikin during the 1950s, prior to the negative solution of the general Burnside problem.

The case of arbitrary exponent has been completely settled in the affirmative by Efim Zelmanov, who was awarded the Fields Medal in 1994 for his work.