A simple construction of infinite finitely generated torsion groups

TL;DR

This paper presents a straightforward proof of Golod's theorem, demonstrating the existence of infinite finitely generated torsion groups using the Nielsen-Schreier index formula, making the proof accessible for educational purposes.

Contribution

It offers a new, simplified proof of Golod's theorem that can be included in standard group theory courses, emphasizing clarity and pedagogical value.

Findings

Existence of infinite finitely generated torsion groups confirmed

Proof based on Nielsen-Schreier index formula

Accessible proof suitable for teaching contexts

Abstract

The goal of this note is to provide yet another proof of the following theorem of Golod: there exists an infinite finitely generated group $G$ such that every element of $G$ has finite order. Our proof is based on the Nielsen-Schreier index formula and is simple enough to be included in a standard group theory course.