The general Burnside problem

Abstract

[EN] The Burnside problems are among the most important problems in group theory in the 20th century. In this project, we will focus on the General Burnside Problem, which asks whether a finitely generated periodic group is necessarily finite, for which the answer is negative.

The notes are organized in three chapters. In the first chapter the reader is introduced to commutator theory, which will be useful to define and work with nilpotent and soluble groups, for which the answer to the General Burnside Problem is affirmative. Then, we also study the problem for linear groups, for which the answer is also affirmative.

In the second and third chapters some negative solutions to the General Burnside Problem are introduced. In the second chapter, Golod-Shafarevich groups are constructed using formal power series and polynomials in non-commuting indeterminates. In the third chapter, we introduce Gupta-Sidki and Grigorchuk groups, using graph theory and automorphisms of trees.