---
title: "Burnside problem"
chunk: 2/3
source: "https://en.wikipedia.org/wiki/Burnside_problem"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T11:02:52.202280+00:00"
instance: "kb-cron"
---

The particular case of B(2, 5) remains open.
The breakthrough in solving the Burnside problem was achieved by Pyotr Novikov and Sergei Adian in 1968. Using a complicated combinatorial argument, they demonstrated that for every odd number n with n > 4381, there exist infinite, finitely generated groups of exponent n. Adian later improved the bound on the odd exponent to 665. In 2015, Adian claimed to have obtained a lower bound of 101 for odd n; however, the full proof of this lower bound was never completed and never published. The case of even exponent turned out to be considerably more difficult. It was only in 1994 that Sergei Vasilievich Ivanov was able to prove an analogue of Novikov–Adian theorem: for any m > 1 and an even n ≥ 248, n divisible by 29, the group B(m, n) is infinite; together with the Novikov–Adian theorem, this implies infiniteness for all m > 1 and n ≥ 248. This was improved in 1996 by I. G. Lysënok to m > 1 and n ≥ 8000. Novikov–Adian, Ivanov and Lysënok established considerably more precise results on the structure of the free Burnside groups. In the case of the odd exponent, all finite subgroups of the free Burnside groups were shown to be cyclic groups. In the even exponent case, each finite subgroup is contained in a product of two dihedral groups, and there exist non-cyclic finite subgroups. 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. First examples of such groups were constructed by A. Yu. Ol'shanskii in 1979 using geometric methods, thus affirmatively solving O. Yu. Schmidt's problem. In 1982 Ol'shanskii was able to strengthen his results to establish existence, for any sufficiently large prime number p (one can take p > 1075) of a finitely generated infinite group in which every nontrivial proper subgroup is a cyclic group of order p. In a paper published in 1996, Ivanov and Ol'shanskii solved an analogue of the Burnside problem in an arbitrary hyperbolic group for sufficiently large exponents.

== Restricted Burnside problem ==
Formulated in the 1930s, it asks another, related, question:

Restricted Burnside problem. If it is known that a group G with m generators and exponent n is finite, can one conclude that the order of G is bounded by some constant depending only on m and n? Equivalently, are there only finitely many finite groups with m generators of exponent n, up to isomorphism?
This variant of the Burnside problem can also be stated in terms of category theory: an affirmative answer for all m is equivalent to saying that the category of finite groups of exponent n has all finite limits and colimits. 
It can also be stated more explicitly in terms of certain universal groups with m generators and exponent n. By basic results of group theory, the intersection of two normal subgroups of finite index in any group is itself a normal subgroup of finite index. Thus, the intersection M of all the normal subgroups of the free Burnside group B(m, n) which have finite index is a normal subgroup of B(m, n). One can therefore define the free restricted Burnside group B0(m, n) to be the quotient group B(m, n)/M. Every finite group of exponent n with m generators is isomorphic to B(m,n)/N where N is a normal subgroup of B(m,n) with finite index. Therefore, by the Third Isomorphism Theorem, every finite group of exponent n with m generators is isomorphic to B0(m,n)/(N/M) — in other words, it is a homomorphic image of B0(m, n).
The restricted Burnside problem then asks whether B0(m, n) is a finite group. 
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. His solution, establishing the finiteness of B0(m, p), used a relation with deep questions about identities in Lie algebras in finite characteristic. 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.

== Notes ==

== References ==