Solution to the Burnside Problem

TL;DR

This paper solves the Burnside Problem for 2-generator groups of prime power exponent, extending the finiteness results to a broader class of groups called GB groups, and discusses potential extensions to more generators.

Contribution

It provides a solution for 2-generator groups of prime power exponent and introduces the Generalized Burnside Theorem applicable to GB groups, including infinite groups.

Findings

Finiteness of 2-generator groups of prime power exponent established

Extension of results to a class of infinite and finite GB groups

Discussion on extending to k-generator groups

Abstract

The Burnside Problem asks whether a finitely generated group of exponent n is finite. We present a solution for 2-generator groups of prime power exponent. Results of P. Hall and G. Higman extends the finiteness conclusion to groups having composite exponents. Our main result, called the Generalized Burnside Theorem, is a solvability theorem that applies to a family of groups called GB (Generalized Burnside) groups that contain infinite as well as finite groups. The final section discusses the extension to k-generator groups although details are left for another time.