Finite Subgroups of Group Rings: A survey

TL;DR

This survey reviews the development of the study of finite subgroups within integral group rings, highlighting key results, methods, and open problems in the field since the 1940s.

Contribution

It provides a comprehensive overview of classical and recent findings, summarizing progress on major problems like the Isomorphism Problem and Zassenhaus Conjectures.

Findings

Resolution of major questions such as the Isomorphism Problem

Progress on the Zassenhaus Conjectures

Identification of new challenging problems

Abstract

In the 1940's Graham Higman initiated the study of finite subgroups of the unit group of an integral group ring. Since then many fascinating aspects of this structure have been discovered. Major questions such as the Isomorphism Problem and the Zassenhaus Conjectures have been settled, leading to many new challenging problems. In this survey we review classical and recent results, sketch methods and list questions relevant for the state of the art.