Finite groups in integral group rings

TL;DR

This paper reviews key problems and techniques related to finite subgroups of units in integral group rings of finite groups, aiming to clarify existing challenges and methods in the field.

Contribution

It provides a comprehensive overview of problems and techniques for studying finite subgroups in integral group rings, consolidating knowledge for researchers and students.

Findings

Summarizes main open problems in the field.

Discusses techniques for analyzing finite subgroups.

Provides insights into future research directions.

Abstract

Notes used for a course held in 2016 in the School of Advances in Group Theory and Applications, for some lectures given in 2018 for the students of the Master in Mathematics of the Vrije Universiteit Brussels, a course for master and Ph.D. students at the Universidade de S\~ao Paulo and at the conference Group algebras, representations and computations. We revise some problems on the study of finite subgroups of the group of units of integral group rings of finite groups and some techniques to attack them.