„Burnside-Problem“ – Versionsunterschied

*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''² = ''b''² = (''ab'')² = 1 together imply that ''ab'' = ''ba'', so that a free Burnside group of exponent two is necessarily [[abelian group|abelian]].</ref>

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> 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 groupsgroup]]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'' > 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.