In 1902 William Burnside [5] wrote 'A still undecided point in the theory of discontinuous groups is whether the order of a group may be not finite, while the order of every operation it contains is finite'. In modern terminology the most general form of the problem is 'can a finitely generated group be infinite while every element in the group has finite order?'. This question was answered in ...

We prove that the free Burnside Group B(3; 2) has order 27 and is isomorphic to ha; b j a3; b3(ab)3; (b 1a)3i. The technique of our proof is also used to show that ha; bja3; b3; a2(ba)nb2i is the semidirect product Cn2+n+1o C3.

This paper introduces two new Burnside rings for a finite group G, called the slice Burnside ring and the section Burnside ring. They are built as Grothendieck rings of the category of morphisms of G-sets, and of Galois morphisms of G-sets, respectively. The well known results on the usual Burnside ring, concerning ghost maps, primitive idempotents, and description of the prime spectrum, are ex...

We prove that every noncyclic subgroup of a free m-generator Burnside group B(m, n) of odd exponent n ≫ 1 contains a subgroup H isomorphic to a free Burnside group B(∞, n) of exponent n and countably infinite rank such that for every normal subgroup K of H the normal closure 〈K〉B(m,n) of K in B(m, n) meets H in K. This implies that every noncyclic subgroup of B(m, n) is SQ-universal in the clas...

It is proved that the free m-generated Burnside groups B(m, n) of exponent n are infinite provided that m > 1 , n > 248 . In 1902 William Burnside posed the following problem [2]. Does a group G have to be finite provided that G has a finite set of generators and its elements satisfy the identity x" — 1 ? In other words, must a finitely generated group G of exponent n be finite? In the same pap...

After many unsuccessful attempts to obtain a proof in the late 30s-early 40s the following weaker version of The Burnside Problem was studied: Is it true that there are only finitely many 7??-generated finite groups of exponent nl In other words the question is whether there exists a universal finite m-generated group of exponent n having all other finite m-generated groups of exponent n as hom...