Editing Burnside problem (section)

== Bounded Burnside problem ==
{{unsolved|mathematics|For which ''m'' and ''n'' is <math>B(m, n)</math> finite?}}
[[File:FreeBurnsideGroupExp3Gens2.png|thumb|350px|right|The [[Cayley graph]] of the 27-element free Burnside group of rank 2 and exponent 3.]]

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'', ''g''<sup>''n''</sup> = 1. A group with this property is said to be ''periodic with bounded exponent'' ''n'', or just a ''group with exponent'' ''n''. The Burnside problem for groups with bounded exponent asks:

<blockquote>'''Burnside problem I.''' If ''G'' is a finitely generated group with exponent ''n'', is ''G'' necessarily finite?</blockquote>

It turns out that this problem can be restated as a question about the finiteness of groups in a particular family. The '''free Burnside group''' of rank ''m'' and exponent ''n'', denoted B(''m'', ''n''), is a group with ''m'' distinguished generators ''x''<sub>1</sub>, ..., ''x<sub>m</sub>'' in which the identity ''x<sup>n</sup>'' = 1 holds for all elements ''x'', and which is the "largest" group satisfying these requirements. More precisely, the characteristic property of B(''m'', ''n'') is that, given any group ''G'' with ''m'' generators ''g''<sub>1</sub>, ..., ''g<sub>m</sub>'' and of exponent ''n'', there is a unique homomorphism from B(''m'', ''n'') to ''G'' that maps the ''i''th generator ''x<sub>i</sub>'' of B(''m'', ''n'') into the ''i''th generator ''g<sub>i</sub>'' of  ''G''. In the language of [[presentation of a group|group presentations]], the free Burnside group B(''m'', ''n'') has ''m'' generators ''x''<sub>1</sub>, ..., ''x<sub>m</sub>'' and the relations ''x<sup>n</sup>'' = 1 for each word ''x'' in ''x''<sub>1</sub>, ..., ''x<sub>m</sub>'', and any group ''G'' with ''m'' generators of exponent ''n'' is obtained from it by imposing additional relations. The existence of the free Burnside group and its uniqueness up to an isomorphism are established by standard techniques of group theory. Thus if ''G'' is any finitely generated group of exponent ''n'', then ''G'' is a [[group homomorphism|homomorphic image]] of B(''m'', ''n''), where ''m'' is the number of generators of ''G''. The Burnside problem for groups with bounded exponent can now be restated as follows:

<blockquote>'''Burnside problem II.''' For which positive integers ''m'', ''n'' is the free Burnside group B(''m'', ''n'') finite?</blockquote>

The full solution to Burnside problem in this form is not known. Burnside considered some easy cases in his original paper:

*B(1, ''n'') is the [[cyclic group]] of order ''n''.
*B(''m'', 2) is the [[direct product of groups|direct product]] of ''m'' copies of the cyclic group of order 2 and hence finite.<ref group="note">The key step is to observe that the identities ''a''<sup>2</sup> = ''b''<sup>2</sup> = (''ab'')<sup>2</sup> = 1 together imply that ''ab'' = ''ba'', so that a free Burnside group of exponent two is necessarily [[abelian group|abelian]].</ref>

The following additional results are known (Burnside, Sanov, [[Marshall Hall (mathematician)|M. Hall]]):

*B(''m'', 3), B(''m'', 4), and B(''m'', 6) are finite for all ''m''.

{{unsolved|mathematics|Is ''B(2, 5)'' finite?}}
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 [[even and odd numbers|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.<ref>[[John Britton (mathematician)|John Britton]] proposed a nearly 300 page alternative proof to the Burnside problem in 1973; however, Adian ultimately pointed out a flaw in that proof.</ref> 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'' ≥ 2<sup>48</sup>, ''n'' divisible by 2<sup>9</sup>, the group B(''m'', ''n'') is infinite; together with the Novikov–Adian theorem, this implies infiniteness for all ''m'' > 1 and ''n'' ≥ 2<sup>48</sup>. 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 group]]s, and there exist non-cyclic finite subgroups. Moreover, the [[word problem for groups|word]] and [[conjugacy problem|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 monster group|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'' > 10<sup>75</sup>) 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.